A375491 - OEIS

%I #13 Aug 18 2024 02:03:31

%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.

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

%H I. 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 , Article 7 (2010).

%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).

%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))

%Y Cf. A000001, A005117.

%K nonn

%O 1,5

%A _Chai Wah Wu_, Aug 17 2024