+ i#v^RIHtHt^RIHt^RIHt^RIHtRRlt RRlt R R lt R R lt R t RtR#))chain combinations)gcd) factorint)as_intc0V^8dQhR\R\/#)factorsreturn)dictbool)formats"_/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/combinatorics/group_numbers.py __annotate__rs$4cVP4FKpVP4F4wr#^p\V4FpWB,V,pV^8XgKR# K6 KM R#)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)r pqem_s& r_is_nilpotent_numberrsU \\^MMODAA1XC!G6  $ rc$V^8dQhR\/#r r r )rs"rrrs..d.rcr\V4pV^8:d\RV,4h\\V44#)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 $n must be a positive integer, not %i)r ValueErrorrr)ns&ris_nilpotent_numberr#s40 q AAv?!CDD  ! --rc$V^8dQhR\/#rr)rs"rrr8sRRDRrc&\V4pV^8:d\RV,4h\V4p\;QJd*RVP 44F 'dK RM RM!RVP 444;'d \ V4#)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 r c3*"TF q^8xK R#5i)N.0rs& r $is_abelian_number..Vs/.1u.FTrr!rallvaluesrr"r s& ris_abelian_numberr28sk4 q AAv?!CDDlG 3/gnn./333/gnn./ / Q Q4H4QQrc$V^8dQhR\/#rr)rs"rrrYsSS4Src&\V4pV^8:d\RV,4h\V4p\;QJd*RVP 44F 'dK RM RM!RVP 444;'d \ V4#)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 r c3*"TF q^8HxK R#5iNr(r)s& rr+#is_cyclic_number..ws0/!Av/r-FTr.r1s& ris_cyclic_numberr9Ysk4 q AAv?!CDDlG 30w~~/03330w~~/0 0 R R5I'5RRrc faa VUau0uFGo\;QJdV3RlV4F 'dK RM RM!V3RlV44'gKESkKI ppW, o ^p\P!V 3Rl\\ S 4^,444pVFp\ V4p^pS V, Fbo\ W%,Uu.uFqwS,^8XgKVNK up4pVSV,^, S^, ,,pV'dKbM W6, pK V#uupiuupi)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 c3:<"TFqS,^8gxK R#5ir6r()r*rrs& rr+"_holder_formula..s(K]Q!]sFTc3<<"TFp\SV4xK R#5i)N)r)r*rMs& rr+r<s"O!<1#5#5s)r/r from_iterablerlenset) prime_factorsrFspowersetpsprodrcr?s &` @r_holder_formularJzs~"LMqSS(K](KSSS(K](K%KMALA A"""OuSVAX"OOH WRA51a%1*QQ56A QTAX1q5) )D4   H M6s!D)D)D)D)D. D. cH\V4pV^8:d\RV,4h\V4p\V4^8XEd\ VP 44^,wr#V^8Xd.R OpV\V48d WC,#V^8Xd.R OpV\V48d WS,#V^8:dV#V^8Xd^#V^8Xd^#V^8XdH^=^V,,^\ V^, ^4,,\ V^, ^4,#V^8Xd^V^,,^'V,,R,^\ V^, ^4,,^ \ V^, ^4,,^\ V^, ^4,,#V^8XEdnV^8XdR#^V^,,^ V^,,,^,V^,,,^V^,,,RV,,R,^V^,,^,V,,R,\ V^, ^4,,V^,^V,,^,\ V^, ^4,,^V,^,\ V^, ^4,,^\ V^, ^4,,^\ V^, ^4,,\ V^, ^ 4,#\;QJd*RVP44F 'gK RM R M!RVP444'da/^ ^b^^b^^b^^b^^b^$^b^(^b^,^b^-^b^0^4b^2^b^4^b^6^b^8^ b^<^ b^?^b^D^b^H^2^K^^L^^P^4^T^^X^ ^Z^ ^\^/CpW9d W`,#\R 4h\V4^8Xd-\VP44wr'Wr,^8Xd^#^#\\VP444#) aNumber of groups of order `n`. In [1]_, ``gnu(n)`` is given, so we follow this notation here as well. Parameters ========== n : Integer ``n`` is a positive integer Returns ======= int : ``gnu(n)`` Raises ====== ValueError Number of groups of order ``n`` is unknown or not implemented. For example, gnu(`2^{11}`) is not yet known. On the other hand, gnu(99) is known to be 2, but this has not yet been implemented in this function. Examples ======== >>> from sympy.combinatorics.group_numbers import groups_count >>> groups_count(3) # There is only one cyclic group of order 3 1 >>> # There are two groups of order 10: the cyclic group and the dihedral group >>> groups_count(10) 2 See Also ======== is_cyclic_number `n` is cyclic iff gnu(n) = 1 References ========== .. [1] John H. Conway, Heiko Dietrich and E.A. O'Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731 .. [2] https://oeis.org/A000001 r iXiii i#c3*"TF q^8xK R#5ir6r(r)s& rr+groups_count..s +*Qq5*r-TFz9Number of groups of order n is unknown or not implemented) r7r7r 3i i ii!lyLZ .) r7r7r rNCii^$imMlNC) rr!rrAlistrranyr0sortedrrJrB)r"r rrA000679A090091smallrs& r groups_countrYsd 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^s6)#+'$.<RBSBM `\0r