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

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

Author

Chai Wah Wu, Aug 17 2024

Keywords

• nonn

References

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

Links

• 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).

Crossrefs

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

Programs

Python

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()))

Formula

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