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Number of nonabelian groups of order m where m is the n-th squarefree number.
1

%I #27 Aug 18 2024 11:42:53

%S 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 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 5,1,0,5,0,1,0,1,0,0,1,3,0,0,1,0,3,0,0

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

%D O. Hölder. Die Gruppen mit quadratfreier Ordnungszahl. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, pages 211-219 (1895).

%H Chai Wah Wu, <a href="/A375483/b375483.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = A060689(A005117(n)) = A375491(n) - A000688(A005117(n)).

%o (Python)

%o from math import isqrt, prod

%o from itertools import combinations

%o from sympy import mobius, primefactors, npartitions, factorint

%o def A375483(n):

%o def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o a = set(primefactors(m))

%o 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 Cf. A000001, A000688, A005117, A060689, A375491.

%K nonn

%O 1,19

%A _Chai Wah Wu_, Aug 17 2024