*h`SrSSKJrJr SSKJr SSKJr SSKJr SSK J r SSK J r SS K Jr SS KJr S r"S S 5r\"5rSr\"SSSS9S5rS\S\4SjrS SjrSrS!SjrSrS"SjrS#SjrSrSrg)$z" Generating and counting primes. )bisect bisect_leftcount)array)randint)sqrt)isprime) deprecated)as_intc0SSKJn [U"U55$)zWrapping ceiling in as_int will raise an error if there was a problem determining whether the expression was exactly an integer or not.r)ceiling)#sympy.functions.elementary.integersrr )ars S/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/ntheory/generate.py_as_int_ceilingrs< '!* cr\rSrSrSrSSjrSrSSjrSrSr S r SS jr S r S r S rSrSrSrSrg)SieveaA list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes. When a lookup is requested involving an odd number that has not been sieved, the sieve is automatically extended up to that number. Implementation details limit the number of primes to ``2^32-1``. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> 25 in sieve False >>> sieve._list array('L', [2, 3, 5, 7, 11, 13, 17, 19, 23]) c0^STl[S/SQ5Tl[S/SQ5Tl[S/SQ5TlUS::a [ S5eUTl[U4S jTRTRTR455(deg ) zInitial parameters for the Sieve class. Parameters ========== sieve_interval (int): Amount of memory to be used Raises ====== ValueError If ``sieve_interval`` is not positive. L) )rr r rri)rr r#rr#rz+sieve_interval should be a positive integerc3T># UHn[U5TR:Hv M g7fN)len_n).0r"selfs r !Sieve.__init__..Cs U.T3q6TWW$.Ts%(N)r'_array_list_tlist_mlist ValueErrorsieve_intervalall)r)r1s` r__init__Sieve.__init__-sC!56 S"45 S"78 Q JK K,Utzz4;; .TUUUUUrc.SS[UR5URSURSURSURSURSS[UR5URSURSURSURSURSS [UR5URSURSURSURSURS4-$) Nzs<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>primerr rr#totientmobius)r&r-r.r/)r)s r__repr__Sieve.__repr__Es6c$**oA 1 tzz!}BB DKK(QQQR$++b/ s4;;'QQQR$++b/ :C C CrNc [SXU455(aS=n=p#U(aURSURUlU(aURSURUlU(aURSURUlgg)zQReset all caches (default). To reset one or more set the desired keyword to True.c3(# UHoSLv M g7fr%)r(r"s rr*Sieve._reset..Vs;":QDy":sTN)r2r-r'r.r/)r)r6r8r9s r_reset Sieve._resetSsw ;56":; ; ;'+ +E +G HTWW-DJ ++htww/DK ++htww/DK rc 4[U5nURSS-nX:agUS-nX1::a:U=R[SURX#55- slX3S-p2X1::aM:U=R[SURX!S-55- slg)zGrow the sieve to cover all primes <= n. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True r#r Nrr)intr-r, _primerange)r)nnumnum2s rextend Sieve.extend_s Fjjnq  7 Avi JJ&d&6&6s&AB BJAgi fS$"2"23A">?? rc ## US-(aUS-nX:a[URX!- S-5nS/U-nURS[UR[ USU--S-55H'n[ US-U-*S-U-X55HnSXF'M M) [ U5HupuU(dMUSU--S-v M USU-- nX:aMgg7f)aGenerate all prime numbers in the range (a, b). Parameters ========== a, b : positive integers assuming the following conditions * a is an even number * 2 < self._list[-1] < a < b < nextprime(self._list[-1])**2 Yields ====== p (int): prime numbers such that ``a < p < b`` Examples ======== >>> from sympy.ntheory.generate import Sieve >>> s = Sieve() >>> s._list[-1] 13 >>> list(s._primerange(18, 31)) [19, 23, 29] rr TFN)minr1r-rr range enumerate)r)rb block_sizeblockptidxs rrDSieve._primerangexs4 q5 FAeT0015Q,?JFZ'EZZ&T!a*n:Lq:P5Q"RS!a%!)  1Q6 FA$EHGT$E*1a#g+/)+ Z AesB+C1CCc[U5n[UR5U:aFUR[ URSS-55 [UR5U:aMEgg)axExtend to include the ith prime number. Parameters ========== i : integer Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend_to_no(9) >>> sieve._list array('L', [2, 3, 5, 7, 11, 13, 17, 19, 23]) Notes ===== The list is extended by 50% if it is too short, so it is likely that it will be longer than requested. r#g?N)r r&r-rHrC)r)r"s r extend_to_noSieve.extend_to_nosM. 1I$**o! KKDJJrNS01 2$**o!rc## Uc[U5nSnO [S[U55n[U5nX:agURU5 UR[ URU5[ URU5ShvN gN7f)aGenerate all prime numbers in the range [2, a) or [a, b). Examples ======== >>> from sympy import sieve, prime All primes less than 19: >>> print([i for i in sieve.primerange(19)]) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> print([i for i in sieve.primerange(7, 19)]) [7, 11, 13, 17] All primes through the 10th prime >>> list(sieve.primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] Nr)rmaxrHr-r)r)rrNs r primerangeSieve.primerangesz0 9"AAAq)*A"A 6  A::k$**a8)$**a8: : :sBB B B c # # [S[U55n[U5n[UR5nX:agX#::a$[ X5HnURUv M gU=R[ S[ X255- sl[ SU5HlnURUnXTS- :XaGX4-S- U-U-n[ XbU5H*nURU==URUU--ss'M, XA:dMhUv Mn [ X25HinURUnXT:Xa:[ XBU5H*nURU==URUU--ss'M, XA:dMXURUv Mk g7f)zGenerate all totient numbers for the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.totientrange(7, 18)]) [6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16] r Nr)rYrr&r.rLr,)r)rrNrEr"ti startindexjs r totientrangeSieve.totientrangesM ?1% & A    6  V1[kk!n$! KK6#uQ{3 3K1a[[[^Q;"#%!)!1A!5J":!4 A$++a.A*==56H!1[[[^7"1^ A$++a.A*==,6++a.( !sC=FA'F.Fc## [S[U55n[U5n[UR5nX:agX#::a$[ X5HnURUv M gU=R[ SS/X#- -5- sl[ SU5HTnURUnX4-S- U-U-n[ XbU5HnURU==U-ss'M XA:dMPUv MV [ X25HJnURUn[ SU-X$5HnURU==U-ss'M XA:dMFUv ML g7f)a&Generate all mobius numbers for the range [a, b). Parameters ========== a : integer First number in range b : integer First number outside of range Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.mobiusrange(7, 18)]) [-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1] r Nr"rr)rYrr&r/rLr,)r)rrNrEr"mir^r_s r mobiusrangeSieve.mobiusranges%& ?1% & A    6  V1[kk!n$! KK6#sAE{3 3K1a[[[^eaiA-1 za0AKKNb(N16H !1[[[^q1ua+AKKNb(N,6H !sC"E (AE  E c[U5n[U5nUS:a[SU-5eXRS:aUR U5 [ URU5nURUS- U:XaX34$X3S-4$)a&Return the indices i, j of the primes that bound n. If n is prime then i == j. Although n can be an expression, if ceiling cannot convert it to an integer then an n error will be raised. Examples ======== >>> from sympy import sieve >>> sieve.search(25) (9, 10) >>> sieve.search(23) (9, 9) rzn should be >= 2 but got: %sr#r )rr r0r-rHr)r)rEtestrNs rsearch Sieve.search1s"q! 1I q5;a?@ @ zz"~  KKN 4::q ! ::a!e  $4K!e8Orc[U5nUS:deUS-S:XaUS:H$URU5up#X#:H$![[4a gf=f)NrFr)r r0AssertionErrorrh)r)rErrNs r __contains__Sieve.__contains__Ns` q A6M6 q5A:6M{{1~v N+  s;A Ac#<# [S5H nXv M g7f)Nr r)r)rEs r__iter__Sieve.__iter__YsqA'Msc[U[5(asURUR5 URb UROSnUS:a [ S5eUR US- URS- UR2$US:a [ S5e[U5nURU5 UR US- $)zReturn the nth prime numberrr zSieve indices start at 1.) isinstanceslicerVstopstart IndexErrorr-stepr )r)rErus r __getitem__Sieve.__getitem__]s a     aff % ww2AGGEqy!!<==::eai 1669: :1u!!<==q A   a ::a!e$ $r)r-r/r'r.r1)i@B)NNNr%)__name__ __module__ __qualname____firstlineno____doc__r3r:r@rHrDrVrZr`rdrhrlrorx__static_attributes__r>rrrrsO$V0 C 0@2' R36":H#)J*X: %rrc [U5nUS:a [S5eU[[R5::a [U$SSKJn SSKJn US:a![RSU-5 [U$Sn[X"U5R5U"U"U55R5--5nXE:a0XE-S- nU"U5R5U:aUnOUS-nXE:aM0[US- U[US- 5- 5$) a  Return the nth prime number, where primes are indexed starting from 1: prime(1) = 2, prime(2) = 3, etc. Parameters ========== nth : int The position of the prime number to return (must be a positive integer). Returns ======= int The nth prime number. Examples ======== >>> from sympy import prime >>> prime(10) 29 >>> prime(1) 2 >>> prime(100000) 1299709 See Also ======== sympy.ntheory.primetest.isprime : Test if a number is prime. primerange : Generate all primes in a given range. primepi : Return the number of primes less than or equal to a given number. References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number_theorem .. [2] https://en.wikipedia.org/wiki/Logarithmic_integral_function .. [3] https://en.wikipedia.org/wiki/Skewes%27_number r z-nth must be a positive integer; prime(1) == 2rlogliir)r r0r&siever-&sympy.functions.elementary.exponentialr'sympy.functions.special.error_functionsrrHrCevalf nextprime_primepi)nthrErrrrNmids rr6r6ssT s A1uHII C Qx::4x QUQx A AQ#c!f+"3"3"55 67A %ul c7==?Q AaA % QUAQ/ 00rzgThe `sympy.ntheory.generate.primepi` has been moved to `sympy.functions.combinatorial.numbers.primepi`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetcSSKJn U"U5$)aRepresents the prime counting function pi(n) = the number of prime numbers less than or equal to n. .. deprecated:: 1.13 The ``primepi`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.primepi` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. Algorithm Description: In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) $\Rightarrow$ phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most $2\sqrt{n}$ values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Examples ======== >>> from sympy import primepi, prime, prevprime, isprime >>> primepi(25) 9 So there are 9 primes less than or equal to 25. Is 25 prime? >>> isprime(25) False It is not. So the first prime less than 25 must be the 9th prime: >>> prevprime(25) == prime(9) True See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime r)primepi)%sympy.functions.combinatorial.numbersr)rE func_primepis rrrspN ?rrEreturnc US:agU[RS::a[RU5S$[U5n[ US-5Vs/sH o"S-S- PM nnS/[ SUS-5Vs/sH o U-S-S- PM sn-nS/US--n[ SUS-S5HnXR(aMX2S- n[ X!S-U5HnSXW'M [ S[ XU--U5S-S5H>nXW(aMX'-nX::aXG==XHU- -ss'M*XG==X0U-U- -ss'M@ [ U[ XU-S- 5S5HnX7==X7U-U- -ss'M M US$s snfs snf)a1Represents the prime counting function pi(n) = the number of prime numbers less than or equal to n. Explanation =========== In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) $\Rightarrow$ phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most $2\sqrt{n}$ values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Parameters ========== n : int rrr#r FrT)rr-rhr rLrK) rElimr"arr1arr2skiprQr_sts rrrsx 1uEKKO||Aq!! q'C"'a. 1.QUqL.D 1 35C!G+<=+>> from sympy import nextprime >>> [(i, nextprime(i)) for i in range(10, 15)] [(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)] >>> nextprime(2, ith=2) # the 2nd prime after 2 5 See Also ======== prevprime : Return the largest prime smaller than n primerange : Generate all primes in a given range rzith should be positiverr r7r#rr!r)rCr r0rr-rhr&r )rEithr"l_nns rrrzswP AAs AAv1221u  QEKKO||A 519s5;;' ';;quqy) ) KKO U[[! !! AqDB w Q 1:: FA Q 1 Q 1:: FA Q F 1:: FA Q 1:: FA Q rc[U5nUS:a [S5eUS:a SSSSSS.U$U[RS::a1[R U5upX:Xa [US- $[U$S US --nX- S::aUS- n[ U5(aU$US -nOUS-n[ U5(aU$US-n[ U5(aU$US -nM0) aReturn the largest prime smaller than n. Notes ===== Potential primes are located at 6*j +/- 1. This property is used during searching. >>> from sympy import prevprime >>> [(i, prevprime(i)) for i in range(10, 15)] [(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)] See Also ======== nextprime : Return the ith prime greater than n primerange : Generates all primes in a given range rzno preceding primesrrr)rr!rrrr#r rr!)rr0rr-rhr )rErurs r prevprimers& A1u.//1uqQ1-a00EKKO||A 61: 8O AqDBv{ F 1::H Q F 1::H Q 1::H Q rNc## UcSUpX:ag[RSnX::a[RX5ShvN gX::a9[R[[RU5SShvN US-nOUS-(aUS-n[ XS-5nX:a[R X5ShvN UnX::ag[ U5nX:aUv OgMNNlN)7f)aoGenerate a list of all prime numbers in the range [2, a), or [a, b). If the range exists in the default sieve, the values will be returned from there; otherwise values will be returned but will not modify the sieve. Examples ======== >>> from sympy import primerange, prime All primes less than 19: >>> list(primerange(19)) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> list(primerange(7, 19)) [7, 11, 13, 17] All primes through the 10th prime >>> list(primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] The Sieve method, primerange, is generally faster but it will occupy more memory as the sieve stores values. The default instance of Sieve, named sieve, can be used: >>> from sympy import sieve >>> list(sieve.primerange(1, 30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] Notes ===== Some famous conjectures about the occurrence of primes in a given range are [1]: - Twin primes: though often not, the following will give 2 primes an infinite number of times: primerange(6*n - 1, 6*n + 2) - Legendre's: the following always yields at least one prime primerange(n**2, (n+1)**2+1) - Bertrand's (proven): there is always a prime in the range primerange(n, 2*n) - Brocard's: there are at least four primes in the range primerange(prime(n)**2, prime(n+1)**2) The average gap between primes is log(n) [2]; the gap between primes can be arbitrarily large since sequences of composite numbers are arbitrarily large, e.g. the numbers in the sequence n! + 2, n! + 3 ... n! + n are all composite. See Also ======== prime : Return the nth prime nextprime : Return the ith prime greater than n prevprime : Return the largest prime smaller than n randprime : Returns a random prime in a given range primorial : Returns the product of primes based on condition Sieve.primerange : return range from already computed primes or extend the sieve to contain the requested range. References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number .. [2] https://primes.utm.edu/notes/gaps.html Nrr#r )rr-rZrrKrDr)rrNlargest_known_primetails rrZrZsV y!1v++b/##A)));;{5;;:;<<< ! # Q Q q* +Dx$$Q--- v aL 5G   * = .s5=C)C#8C)8C%9AC)=C'>&C)%C)'C)cX:ag[[X45up[US- U5n[U5nX1:a [ U5nX0:a [ S5eU$)aReturn a random prime number in the range [a, b). Bertrand's postulate assures that randprime(a, 2*a) will always succeed for a > 1. Note that due to implementation difficulties, the prime numbers chosen are not uniformly random. For example, there are two primes in the range [112, 128), ``113`` and ``127``, but ``randprime(112, 128)`` returns ``127`` with a probability of 15/17. Examples ======== >>> from sympy import randprime, isprime >>> randprime(1, 30) #doctest: +SKIP 13 >>> isprime(randprime(1, 30)) True See Also ======== primerange : Generate all primes in a given range References ========== .. [1] https://en.wikipedia.org/wiki/Bertrand's_postulate Nr z&no primes exist in the specified range)maprCrrrr0)rrNrErQs r randprimeresZ@ v sQF DAAqA! Av aLuABB HrcU(a [U5nO [U5nUS:a [S5eSnU(a&[SUS-5HnU[ U5-nM U$[ SUS-5HnX#-nM U$)a Returns the product of the first n primes (default) or the primes less than or equal to n (when ``nth=False``). Examples ======== >>> from sympy.ntheory.generate import primorial, primerange >>> from sympy import factorint, Mul, primefactors, sqrt >>> primorial(4) # the first 4 primes are 2, 3, 5, 7 210 >>> primorial(4, nth=False) # primes <= 4 are 2 and 3 6 >>> primorial(1) 2 >>> primorial(1, nth=False) 1 >>> primorial(sqrt(101), nth=False) 210 One can argue that the primes are infinite since if you take a set of primes and multiply them together (e.g. the primorial) and then add or subtract 1, the result cannot be divided by any of the original factors, hence either 1 or more new primes must divide this product of primes. In this case, the number itself is a new prime: >>> factorint(primorial(4) + 1) {211: 1} In this case two new primes are the factors: >>> factorint(primorial(4) - 1) {11: 1, 19: 1} Here, some primes smaller and larger than the primes multiplied together are obtained: >>> p = list(primerange(10, 20)) >>> sorted(set(primefactors(Mul(*p) + 1)).difference(set(p))) [2, 5, 31, 149] See Also ======== primerange : Generate all primes in a given range r zprimorial argument must be >= 1r)r rCr0rLr6rZ)rErrQr"s r primorialrsd 1I F1u:;; A q!a%A qMA! HAq1u%A FA& Hrc## [U=(d S5nS=pEX"U5pvSnU(aUv Xg:waLU(aX:a@US- nXE:Xa UnUS-nSnU(aUv U"U5nUS- nXg:waU(dM9X:aM@U(aX:XaU(agUS4v gU(dFSn U=pg[U5H nU"U5nM Xg:waU"U5nU"U5nU S- n Xg:waMXY4v gg7f)aFor a given iterated sequence, return a generator that gives the length of the iterated cycle (lambda) and the length of terms before the cycle begins (mu); if ``values`` is True then the terms of the sequence will be returned instead. The sequence is started with value ``x0``. Note: more than the first lambda + mu terms may be returned and this is the cost of cycle detection with Brent's method; there are, however, generally less terms calculated than would have been calculated if the proper ending point were determined, e.g. by using Floyd's method. >>> from sympy.ntheory.generate import cycle_length This will yield successive values of i <-- func(i): >>> def gen(func, i): ... while 1: ... yield i ... i = func(i) ... A function is defined: >>> func = lambda i: (i**2 + 1) % 51 and given a seed of 4 and the mu and lambda terms calculated: >>> next(cycle_length(func, 4)) (6, 3) We can see what is meant by looking at the output: >>> iter = cycle_length(func, 4, values=True) >>> list(iter) [4, 17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] There are 6 repeating values after the first 3. If a sequence is suspected of being longer than you might wish, ``nmax`` can be used to exit early (and mu will be returned as None): >>> next(cycle_length(func, 4, nmax = 4)) (4, None) >>> list(cycle_length(func, 4, nmax = 4, values=True)) [4, 17, 35, 2] Code modified from: https://en.wikipedia.org/wiki/Cycle_detection. rr rN)rCrL) fx0nmaxvaluespowerlamtortoiseharer"mus r cycle_lengthrsf tyq>DOE2d A   DAH Q <H QJEC Jw q  DDAH   *    sAT7D{HT7D !GBg  sA5C)9C)A C)"C)c [U5nUS:a [S5e/SQnUS::aX!S- $S[RSpCX[ U5- S- ::aJX4S- :a+X4-S- nU[ U5- S- U:aUnOUnX4S- :aM+[ U5(aUS-nU$SSKJn SS KJ n Sn[X"U5U"U"U55--5nX4:a'X4-S- nXW"U5- S- U:aUnOUS-nX4:aM'U[ U5- S- nX:a![ U5(dUS-nUS-nX:aM![ U5(aUS-nU$) aReturn the nth composite number, with the composite numbers indexed as composite(1) = 4, composite(2) = 6, etc.... Examples ======== >>> from sympy import composite >>> composite(36) 52 >>> composite(1) 4 >>> composite(17737) 20000 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n prime : Return the nth prime compositepi : Return the number of positive composite numbers less than or equal to n r z1nth must be a positive integer; composite(1) == 4) r!rr rr!r#rrr) r r0rr-rr rrrrrC) rrE composite_arrrrNrrr n_compositess r compositer+so0 s A1uLMM8MBwU## ekk"oq Oa a%i5Q,CXc]"Q&* a%i 1:: FA:: A As1vCF # $%A %ul C=1 q AaA %x{?Q&L  qzz A L Q  qzz Q HrcH[U5nUS:agU[U5- S- $)aReturn the number of positive composite numbers less than or equal to n. The first positive composite is 4, i.e. compositepi(4) = 1. Examples ======== >>> from sympy import compositepi >>> compositepi(25) 15 >>> compositepi(1000) 831 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime primepi : Return the number of primes less than or equal to n composite : Return the nth composite number r!rr )rCr)rEs r compositepirls*, AA1u x{?Q r)r r%)T)NF) r~rr itertoolsrrr,sympy.core.randomrsympy.external.gmpyr primetestr sympy.utilities.decoratorr sympy.utilities.miscr rrrr6rrCrrrrZrrrrrr>rrrs '"%$0'T%T%n F1R  kBDU DUp_s_s_DPf,^fR) X? DUp> Br