A375491 -id:A375491 - cp's OEIS frontend

A384185 Integers in A375491 in order of their first appearance.

1, 2, 4, 6, 12, 5, 8, 24, 3, 7, 18, 16, 30, 36, 48, 10, 32, 14, 72, 9, 60, 96, 120, 19, 64, 13, 40, 144, 21, 35, 56, 38, 28, 90, 26, 240, 15, 192, 384, 44, 76, 360, 88, 80, 180, 168, 114, 54, 288, 112, 264, 25, 41, 33, 256, 98, 20, 55, 312, 128

Offset: 1

Robin Jones, May 21 2025

nonn

Does every positive integer appear in this sequence?

			a(1)=1 since the first squarefree integer (1) has 1 group of that order.
a(2)=2 since the first squarefree integer not to have a(1)=1 group of its order (6) has 2 groups of its order.
a(3)=4 since the first squarefree integer not to have a(1)=1 or a(2)=2 groups of its order (30) has 4 groups of its order.

Cf. A375491.

A375483 Number of nonabelian groups of order m where m is the n-th squarefree number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 3, 0, 0, 1, 0, 0, 1, 1, 0, 5, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 1, 0, 5, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 3, 0, 1, 1, 0, 0, 5, 1, 0, 5, 0, 1, 0, 1, 0, 0, 1, 3, 0, 0, 1, 0, 3, 0, 0

Offset: 1

Chai Wah Wu, Aug 17 2024

nonn

O. Hölder. Die Gruppen mit quadratfreier Ordnungszahl. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, pages 211-219 (1895).

Chai Wah Wu, Table of n, a(n) for n = 1..10000
I. Ganev, Groups of a Square-Free order, Rose-Hulman Undergraduate Mathematics Journal: Vol. 11 : Iss. 1 , Article 7 (2010).

Cf. A000001, A000688, A005117, A060689, A375491.

from math import isqrt, prod
from itertools import combinations
from sympy import mobius, primefactors, npartitions, factorint
def A375483(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))-prod(map(npartitions, factorint(m).values()))

a(n) = A060689(A005117(n)) = A375491(n) - A000688(A005117(n)).

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A384185 Integers in A375491 in order of their first appearance.

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A375483 Number of nonabelian groups of order m where m is the n-th squarefree number.

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