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A375483 Number of nonabelian groups of order m where m is the n-th squarefree number.

%I A375483 #27 Aug 18 2024 11:42:53
%S A375483 0,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,1,0,3,0,0,1,0,0,1,1,0,5,0,1,0,0,0,1,
%T A375483 1,1,0,0,1,0,3,0,0,3,0,0,1,0,5,0,1,0,0,1,0,0,0,1,1,0,0,0,3,0,1,1,0,0,
%U A375483 5,1,0,5,0,1,0,1,0,0,1,3,0,0,1,0,3,0,0
%N A375483 Number of nonabelian groups of order m where m is the n-th squarefree number.
%D A375483 O. Hölder. Die Gruppen mit quadratfreier Ordnungszahl. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, pages 211-219 (1895).
%H A375483 Chai Wah Wu, <a href="/A375483/b375483.txt">Table of n, a(n) for n = 1..10000</a>
%H A375483 I. Ganev, <a href="https://scholar.rose-hulman.edu/rhumj/vol11/iss1/7">Groups of a Square-Free order</a>, Rose-Hulman Undergraduate Mathematics Journal: Vol. 11 : Iss. 1 , Article 7 (2010).
%F A375483 a(n) = A060689(A005117(n)) = A375491(n) - A000688(A005117(n)).
%o A375483 (Python)
%o A375483 from math import isqrt, prod
%o A375483 from itertools import combinations
%o A375483 from sympy import mobius, primefactors, npartitions, factorint
%o A375483 def A375483(n):
%o A375483     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A375483     m, k = n, f(n)
%o A375483     while m != k:
%o A375483         m, k = k, f(k)
%o A375483     a = set(primefactors(m))
%o A375483     return sum(prod((p**sum(1 for q in b if q%p==1)-1)//(p-1) for p in a-set(b)) for l in range(0,len(a)+1) for b in combinations(a,l))-prod(map(npartitions, factorint(m).values()))
%Y A375483 Cf. A000001, A000688, A005117, A060689, A375491.
%K A375483 nonn
%O A375483 1,19
%A A375483 _Chai Wah Wu_, Aug 17 2024