\h#SSKJrJr SSKJr SSKJr SSKJr S\ S\ 4Sjr S\ 4Sjr S\ 4S jr S\ 4S jrS rS rg ))chain combinations)gcd) factorint)as_intfactorsreturncUR5HAnUR5H*up#Sn[U5HnXB-U-nUS:XdM g M, MC g)zCheck whether `n` is a nilpotent number. Note that ``factors`` is a prime factorization of `n`. This is a low-level helper for ``is_nilpotent_number``, for internal use. FT)keysitemsrange)rpqem_s Y/var/www/auris/envauris/lib/python3.13/site-packages/sympy/combinatorics/group_numbers.py_is_nilpotent_numberrsU \\^MMODAA1XC!G6  $ ch[U5nUS::a[SU-5e[[U55$)a& Check whether `n` is a nilpotent number. A number `n` is said to be nilpotent if and only if every finite group of order `n` is nilpotent. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_nilpotent_number >>> from sympy import randprime >>> is_nilpotent_number(21) False >>> is_nilpotent_number(randprime(1, 30)**12) True References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A056867 r$n must be a positive integer, not %i)r ValueErrorrr)ns ris_nilpotent_numberrs40 q AAv?!CDD  ! --rc[U5nUS::a[SU-5e[U5n[SUR 555=(a [ U5$)a: Check whether `n` is an abelian number. A number `n` is said to be abelian if and only if every finite group of order `n` is abelian. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_abelian_number >>> from sympy import randprime >>> is_abelian_number(4) True >>> is_abelian_number(randprime(1, 2000)**2) True >>> is_abelian_number(60) False References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A051532 rrc3*# UH oS:v M g7f)N.0rs r $is_abelian_number..Vs/.1u.rrrallvaluesrrrs ris_abelian_numberr)8sQ4 q AAv?!CDDlG /gnn./ / Q4H4QQrc[U5nUS::a[SU-5e[U5n[SUR 555=(a [ U5$)a3 Check whether `n` is a cyclic number. A number `n` is said to be cyclic if and only if every finite group of order `n` is cyclic. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_cyclic_number >>> from sympy import randprime >>> is_cyclic_number(15) True >>> is_cyclic_number(randprime(1, 2000)**2) False >>> is_cyclic_number(4) False References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A003277 rrc3*# UH oS:Hv M g7fr Nrr s rr"#is_cyclic_number..ws0/!Av/r$r%r(s ris_cyclic_numberr.YsQ4 q AAv?!CDDlG 0w~~/0 0 R5I'5RRrc ^^ UV^s1sH!m[U4SjU55(dMTiM# nnX- m Sn[R"U 4Sj[[ T 5S-555nUHbn[ U5nSnT U- HEm[ X%-Vs/sHowT-S:XdM UPM sn5nUTU-S- TS- --nU(aME O X6- nMd U$s snfs snf)aNumber of groups of order `n`. where `n` is squarefree and its prime factors are ``prime_factors``. i.e., ``n == math.prod(prime_factors)`` Explanation =========== When `n` is squarefree, the number of groups of order `n` is expressed by .. math :: \sum_{d \mid n} \prod_p \frac{p^{c(p, d)} - 1}{p - 1} where `n=de`, `p` is the prime factor of `e`, and `c(p, d)` is the number of prime factors `q` of `d` such that `q \equiv 1 \pmod{p}` [2]_. The formula is elegant, but can be improved when implemented as an algorithm. Since `n` is assumed to be squarefree, the divisor `d` of `n` can be identified with the power set of prime factors. We let `N` be the set of prime factors of `n`. `F = \{p \in N : \forall q \in N, q \not\equiv 1 \pmod{p} \}, M = N \setminus F`, we have the following. .. math :: \sum_{d \in 2^{M}} \prod_{p \in M \setminus d} \frac{p^{c(p, F \cup d)} - 1}{p - 1} Practically, many prime factors are expected to be members of `F`, thus reducing computation time. Parameters ========== prime_factors : set The set of prime factors of ``n``. where `n` is squarefree. Returns ======= int : Number of groups of order ``n`` Examples ======== >>> from sympy.combinatorics.group_numbers import _holder_formula >>> _holder_formula({2}) # n = 2 1 >>> _holder_formula({2, 3}) # n = 2*3 = 6 2 See Also ======== groups_count References ========== .. [1] Otto Holder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893). http://dx.doi.org/10.1007/BF01443651 .. [2] John H. Conway, Heiko Dietrich and E.A. O'Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731 c32># UH oT-S:gv M g7fr,r)r!rrs rr""_holder_formula..s(K]Q!]src3<># UHn[TU5v M g7f)N)r)r!rMs rr"r1s"O!<1#5#5sr )r&r from_iterablerlenset) prime_factorsrFspowersetpsprodrcr4s ` @r_holder_formular?zs~"LMqS(K](K%KMALA A"""OuSVAX"OOH WRA51a%1*Q56A QTAX1q5) )D4   H M6sCC C "C c [U5nUS::a[SU-5e[U5n[U5S:XGa[ UR 55Sup#US:Xa/SQnU[U5:aXC$US:Xa/SQnU[U5:aXS$US::aU$US:XagUS :Xag US:Xa+S SU--S[ US- S5--[ US- S 5-$US :XaJSUS--S U--S-S[ US- S5--S[ US- S 5--S[ US- S5--$US:XaUS:XagSUS--SUS ---SUS---SUS---SU--S-S US--SU--S-[ US- S5--US-SU--S-[ US- S 5--SU-S-[ US- S5--S [ US- S5--S[ US- S5--[ US- S5-$[SUR555(aT0SS_SS_S S_SS _S!S _S"S#_S$S#_SS _S%S_S&S'_S(S_S'S_S)S _S*S+_S,S+_S-S _S.S_S(SS S'S SS/S S0.EnX;aX`$[S15e[U5S:Xa'[UR55up'Xr-S:XaS$S$[[UR555$)2aNumber of groups of order `n`. In [1]_, ``gnu(n)`` is given, so we follow this notation here as well. Parameters ========== n : Integer ``n`` is a positive integer Returns ======= int : ``gnu(n)`` Raises ====== ValueError Number of groups of order ``n`` is unknown or not implemented. For example, gnu(`2^{11}`) is not yet known. On the other hand, gnu(99) is known to be 2, but this has not yet been implemented in this function. Examples ======== >>> from sympy.combinatorics.group_numbers import groups_count >>> groups_count(3) # There is only one cyclic group of order 3 1 >>> # There are two groups of order 10: the cyclic group and the dihedral group >>> groups_count(10) 2 See Also ======== is_cyclic_number `n` is cyclic iff gnu(n) = 1 References ========== .. [1] John H. Conway, Heiko Dietrich and E.A. O'Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731 .. [2] https://oeis.org/A000001 rrr ) r r rA3i i ii!lyLZ .r) r r rArBCii^$imMlNCrBrE='iX i ,ii i# c3*# UH oS:v M g7fr,rr s rr"groups_count..s +*Qq5*r$$rC(-04268 <?D )HKLPTXZ\z9Number of groups of order n is unknown or not implemented) rrrr6listr ranyr'sortedr r?r7)rrrrA000679A090091smallrs r groups_countrvsd q AAv?!CDDlG 7|qgmmo&q) 6SG3w<z! 6JG3w<z! 6H 6 6 6!8aAaC m+c!A#qk9 9 6QT6BqD=3&S1a[.!#%c!A#qk>245c!A#qkMB B 6AvQT6Bq!tG#bAg-AqD83q5@4GQT6BqD=3&AaC 3478!tbd{S7H#aPQcST+6UVQ38S1a[()+,S1a[=9;r}s[)#+'$4$.d.<RDRBS4SBM `\0r