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

1,5

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.

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).

Mathematica

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

Prog

(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))
(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

Crossrefs

Cf. A000001, A005117.

Sequence in context: A249615 A096860 A128185 * A175244 A206722 A245222

Adjacent sequences: A375488 A375489 A375490 * A375492 A375493 A375494

Keyword

nonn

Author

Chai Wah Wu, Aug 17 2024

Status

approved