A375491 - OEIS

%I #30 Aug 14 2025 01:10:41

%S 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 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 6,2,1,6,1,2,1,2,1,1,2,4,1,1,2,1,4,1,1

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

%H Chai Wah Wu, <a href="/A375491/b375491.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 a(n) = A000001(A005117(n)).

%F 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 FiniteGroupCount[Select[Range[150], SquareFreeQ]] (* _Amiram Eldar_, Jul 13 2025 *)

%o (Python)

%o from math import isqrt, prod

%o from itertools import combinations

%o from sympy import mobius, primefactors

%o def A375491(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))

%o (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 Cf. A000001, A005117.

%K nonn

%O 1,5

%A _Chai Wah Wu_, Aug 17 2024