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

Author

Eric M. Schmidt, Sep 15 2014

Keywords

• nonn

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.

Crossrefs

Cf. A000001, A069209.

Programs

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)))

Formula

a(A005117(n)) = A000001(A005117(n)). - Michel Marcus, Sep 15 2014