\h##SSKJrJrJrJrJrJrJrJrJ r J r J r J r J r JrJrJr SSKJrJr SSKJr SSKJr SSKrSSKJr SrS rS r\"SS/5S \S \4S j5rS \S \4Sjr \"SSSS9SSj5r!g))fzerofrom_int from_rationalfonefhalfbitcountto_intmpf_mulmpf_divmpf_submpf_addmpf_sqrtmpf_pi mpf_cosh_sinhmpf_cosmpf_sin)_sqrt_mod_prime_poweris_quad_residue) deprecated)recurrence_memoNcountcSnS/U-qS/U-q[US-5S-n[SU5H?n[US:XdM[X"-X5Hn[US:XdMU[U'M MA [SU5Hgn[US:XaU[U'US- [U'M'[UnX$-nXT-S:Xa[UU-[U'MQ[UUS- -[U'Mi g)N順rr?)_factor_totientintrange)maxnlimijxys Q/var/www/auris/envauris/lib/python3.13/site-packages/sympy/ntheory/partitions_.py_prer* s Dc$hGs4xH dCi.1 C 1c] 1:?13(1:?!"GAJ) 1d^ 1:?GAJA#HQK  AJ D 5A:"1+a-HQK"1+q1u-HQKc US:Xa[$UnSn[UnX5-S:XaX5-nUS- nX5-S:XaMX-nSSU-- n[U5nUS:XGawUS:XaSU-n XU --nU[SUS- U 5-U -n[ USUS-5Sn [ [ [SU -5X5[U 5U5n [ [ [S U-SU S-- -5[[U5U5U5[X5U5$US:XaSU-n XU --nUS:aU[S US-S- U 5-U -n[ USUS-5Sn [ [ [SU -5X5[U 5U5n [ [ [SS US---SSU S--- -5[[US-5U5U5[X5U5$XU--nUS -S ;aS OSn Xu-S:Xa5US:Xa)[ [U 5[[U5U5U5$[$[Xu5(d[$XTS- -US- -n U[S U S- U5-n[ XuU5Sn [ [ [SU -5X5[U5U5n [ [ [SU -5[[U5U5U5[X5U5$US:wdUS:a[R"US5[R"US5pSX--nX-U-US-S- U--[XF-U-U-[US- U5-U-nX-U-US-S- U--[XC-U-U-[US- U5-U-n[ [!UX25[!UXb5U5$US:XawSU-S-[S[US- U5-U-nSUS- US-S- S-- US--S--S-n[ [ [S 5[!UX25U5[!UXb55$SU-S-[S[US- U5-U-nSXS-S- S-- S--S-n[ [!UX25[!UXb5U5$)z{Compute the inner sum in HRR formula [1]_ References ========== .. [1] https://msp.org/pjm/1956/6-1/pjm-v6-n1-p18-p.pdf rrr @ )ri@r )rrrpowrr r rrrrrrmathgcdr _a)nkpreck1epk2vpimodmargjacobi3_phid1d2n1n2s r)r;r;"s Av B A A &A+  Q &A+ B BqDA B Qw 6A#C#g A3q!a%%%,A%aAE215A'1 r)*23-?C7"q!q1u+./!d+T3"D* * 6A#C#g A1us2q!tax--4%aAE215A'(1Q3-: t%C7B!a%=!aQi-89!Q$.6"D* * AIB&("a 5A:AvW%Xa[$/77Lq$$Lq5z1q5! s3q!$ $ !! *1 - HQqSM2 , QKw QwY  Xa[$ '/ C & &  Ava"b!488B#3B KtAvQR' R HRL1,b 12578tAvQR' R HRL1,b 12578r"b'B)94@@AvsQwC"!1266" <AERUQYN*RU3q88A =w RL r2 d$ r2   Q37CHRL1,b1 1R 7B qEAI>!Q& &! +B 2b"#RB%5t <sq23pisqrt8rDabcchshDEs r)_drXys  A B4 A ]1b$7>AA 71=$ /FB!2& .A t 4d;A 1=r+r<returncSnSn[5HDnUSU-S-- nX- nUS:a U$XnXTS--nSU::aXaU- nX$S-(aU*OU- nMF U$)z_Calculate the partition function P(n) Parameters ========== n : int nonnegative integer rr0rrr)r<prevrCpentar%npss r)_partition_recr_s A E W 1q Y 6  H H !e  7 bMA q5aRa Hr+c ^^^US:agUS::aU[R5- S:d[R5S:XaUS:a [U5$S[5;a [5 [ [ R SU-S- S--[ R"S U-5- [ R"S 5- S -5[ R"S 5-n[ US -S -5=p#S[ R S--S[ R"S5-- m[ R [ R"S5-S- m[ R [ R"S5-mUUU4Sjn[S[ R"US-55nU"X5S:deUS:a%U"X5S:aUS-nUS:aU"X5S:aMUnUS-nXV- S :a*XV-S-nU"X5=nS:aUnOUS:aUnXV- S :aM*Un U S:a [S5e[n [[[!SSU5U5[#U5U5n [[%S5U5n ['S U 5HRn [)X U5n[+X X;U 5n[-U [X5U5n [/[1[3U555S-nMT [ [3[-U [4U555$)zGCalculate the partition function P(n) Parameters ========== n : int ri@ Fri@8rg@rr1 rg?d,r0KgUUUUUU?c>[RnTU"U5- TU"XS- - 5-[R"TU"U5-U- 5--$)Nr)r9sqrtsinh)r<Nrhc1c2c3s r)_M_partition.._MsHyy$q'zBtA1uI.tyyDGA/FFFFr+r/(rz Input too bigr.2)r_ cache_lengthglobalsr*r!r9rDloglog2rhmaxceil ValueErrorrr rrrrr"r;rXr rabsr r)r<pbitsr>rArnbigsmallrjerMr^rOrPqrQdrkrlrms @@@r) _partitionrs 1u W ^88::R?  ' ' )Q .1v:a   !   1R#  1  xx|$&'( ) "  E59s?##D DGGQJDIIaL( )B 1 b B 3 BG a1c6" #C a:   (r!zC'   (r!zC' E 'C +/ [1 Q(NBc !C 3YE +/ A 5y)) A XmAq!4a8&)Q GF Xa[! $E 1a[ qQK qQ & Awq}d + S^ $r ) vga-. //r+zpThe `sympy.ntheory.partitions_.npartitions` has been moved to `sympy.functions.combinatorial.numbers.partition`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetcSSKJn U"U5$)a Calculate the partition function P(n), i.e. the number of ways that n can be written as a sum of positive integers. .. deprecated:: 1.13 The ``npartitions`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.partition` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. P(n) is computed using the Hardy-Ramanujan-Rademacher formula [1]_. The correctness of this implementation has been tested through $10^{10}$. Examples ======== >>> from sympy.functions.combinatorial.numbers import partition >>> partition(25) 1958 References ========== .. [1] https://mathworld.wolfram.com/PartitionFunctionP.html r) partition)%sympy.functions.combinatorial.numbersr)r<verbosefunc_partitions r) npartitionsrsBR ! r+)F)" mpmath.libmprrrrrrr r r r r rrrrrresidue_ntheoryrrsympy.utilities.decoratorrsympy.utilities.memoizationrr9 itertoolsrr*r;rXr!r_rrr+r)rs@@@@@D07 ..U=n$!Q c C  4J0#J0#J0Z  tBD Dr+