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A375491
Number of groups of order m where m is the n-th squarefree number.
2
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
REFERENCES
O. Hölder. Die Gruppen mit quadratfreier Ordnungszahl. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, pages 211-219 (1895).
LINKS
I. Ganev, Groups of a Square-Free order, Rose-Hulman Undergraduate Mathematics Journal: Vol. 11 : Iss. 1 , Article 7 (2010).
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).
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))
CROSSREFS
Sequence in context: A249615 A096860 A128185 * A175244 A206722 A245222
KEYWORD
nonn
AUTHOR
Chai Wah Wu, Aug 17 2024
STATUS
approved