A247371 Number of groups of order n for which all Sylow subgroups are cyclic.

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

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

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

Cf. A000001, A069209.

Sequence in context: A247462 A323172 A327403 * A331177 A173751 A126864

Adjacent sequences: A247368 A247369 A247370 * A247372 A247373 A247374

Keyword

nonn

Author

Eric M. Schmidt, Sep 15 2014

Status

approved