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

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, 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, 6, 2, 1, 6, 1, 2, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 4, 1, 1

Offset: 1

Author

Chai Wah Wu, Aug 17 2024

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Iordan Ganev, Groups of a Square-Free Order, Rose-Hulman Undergraduate Mathematics Journal, Vol. 11, Iss. 1 (2010), Article 7.
• Otto Hölder, Die Gruppen mit quadratfreier Ordnungszahl, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1895), pages 211-229.

Crossrefs

Cf. A000001, A005117.

Programs

Mathematica

FiniteGroupCount[Select[Range[150], SquareFreeQ]] (* Amiram Eldar, Jul 13 2025 *)

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

Python

from math import isqrt, prod
from itertools import combinations
from sympy import mobius, primefactors
def A375491(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    a = set(primefactors(m))
    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))

Formula

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

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).