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A375491 Number of groups of order m where m is the n-th squarefree number.

%I A375491 #30 Aug 14 2025 01:10:41
%S A375491 1,1,1,1,2,1,2,1,1,2,1,1,1,2,2,1,2,1,4,1,1,2,1,1,2,2,1,6,1,2,1,1,1,2,
%T A375491 2,2,1,1,2,1,4,1,1,4,1,1,2,1,6,1,2,1,1,2,1,1,1,2,2,1,1,1,4,1,2,2,1,1,
%U A375491 6,2,1,6,1,2,1,2,1,1,2,4,1,1,2,1,4,1,1
%N A375491 Number of groups of order m where m is the n-th squarefree number.
%H A375491 Chai Wah Wu, <a href="/A375491/b375491.txt">Table of n, a(n) for n = 1..10000</a>
%H A375491 Iordan Ganev, <a href="https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1121&amp;context=rhumj">Groups of a Square-Free Order</a>, Rose-Hulman Undergraduate Mathematics Journal, Vol. 11, Iss. 1 (2010), Article 7.
%H A375491 Otto Hölder, <a href="http://dml.mathdoc.fr/item/GDZPPN002497018/">Die Gruppen mit quadratfreier Ordnungszahl</a>, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1895), pages 211-229.
%F A375491 a(n) = A000001(A005117(n)).
%F A375491 a(n) = Sum_{d|m} Product_{p} (p^c(p)-1)/(p-1) where m is the n-th squarefree number and p is a prime factor of m/d and c(p) is the number of prime factors of d that are congruent to 1 modulo p (Hölder).
%t A375491 FiniteGroupCount[Select[Range[150], SquareFreeQ]] (* _Amiram Eldar_, Jul 13 2025 *)
%o A375491 (Python)
%o A375491 from math import isqrt, prod
%o A375491 from itertools import combinations
%o A375491 from sympy import mobius, primefactors
%o A375491 def A375491(n):
%o A375491     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A375491     m, k = n, f(n)
%o A375491     while m != k:
%o A375491         m, k = k, f(k)
%o A375491     a = set(primefactors(m))
%o A375491     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))
%o A375491 (PARI) apply( {A375491(n, m=A005117(n))=sumdiv(m, d, my(f=factor(d)[,1]); vecprod([ (p^vecsum([q%p==1| q<-f])-1)/(p-1) | p<-factor(m/d)[,1] ]))}, [1..66]) \\ _M. F. Hasler_, Aug 08 2025
%Y A375491 Cf. A000001, A005117.
%K A375491 nonn
%O A375491 1,5
%A A375491 _Chai Wah Wu_, Aug 17 2024