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A247371
Number of groups of order n for which all Sylow subgroups are cyclic.
2
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 1, 4, 1, 3, 1, 4, 1, 2, 1, 2, 1, 2, 1, 6, 1, 3, 1, 2, 1, 6, 1, 2, 1
OFFSET
1,6
COMMENTS
For squarefree n this gives the total number of groups of order n.
LINKS
M. Ram Murty and V. Kumar Murty, On groups of squarefree order, Math. Ann. 267, no. 3, 299-309, 1984.
FORMULA
a(A005117(n)) = A000001(A005117(n)). - Michel Marcus, Sep 15 2014
PROG
(Sage)
def pnu(pp, m) : return prod(gcd(pp, q-1) for q in prime_divisors(m))
def a(n) : s = n.radical(); return sum(prod(sum((pnu(p^(k+1), s//prod(c)) - pnu(p^k, s//prod(c))) // (p^k*(p-1)) for k in range(n.valuation(p))) for p in c) for c in powerset(prime_divisors(n)))
CROSSREFS
Sequence in context: A247462 A323172 A327403 * A331177 A173751 A126864
KEYWORD
nonn
AUTHOR
Eric M. Schmidt, Sep 15 2014
STATUS
approved