

A069626


Number of distinct sets of numbers whose least common multiple is n.


3



1, 1, 1, 2, 1, 5, 1, 4, 2, 5, 1, 22, 1, 5, 5, 8, 1, 22, 1, 22, 5, 5, 1, 92, 2, 5, 4, 22, 1, 109, 1, 16, 5, 5, 5, 200, 1, 5, 5, 92, 1, 109, 1, 22, 22, 5, 1, 376, 2, 22, 5, 22, 1, 92, 5, 92, 5, 5, 1, 1874, 1, 5, 22, 32, 5, 109, 1, 22, 5, 109, 1, 1696, 1, 5, 22, 22, 5, 109, 1, 376, 8, 5, 1, 1874, 5, 5, 5, 92, 1, 1874, 5, 22
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OFFSET

1,4


COMMENTS

(1,n) counts as one such set and 1 may not occur in any other set.
a(p) = 1, a(p*q) = 5, a(p^2*q) = 13, a(p^3) = 4, a(p^4) = 8 etc. where p and q are primes. It can be shown that a(p^k) = 2^(k1). Problem: find an expression for a(N) when N = p^a*q^b*r^c*..., p,q,r are primes.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000


FORMULA

a(n) = Sum_{ d divides n } mu(n/d)*2^(tau(d)1).  Vladeta Jovovic, Jul 07 2003


EXAMPLE

a(6) = 5 as the five distinct sets are (1, 6), (2, 6), (3, 6), (2, 3) and (2, 3, 6).
a(12) = 22 from (1,12), (4,3), (2,4,3), (4,6), (2,4,6), (4,3,6), (2,4,3,6), (2,12), (4,12), (2,4,12), (3,12), (2,3,12), (4,3,12), (2,4,3,12), (6,12), (2,6,12), (4,6,12), (2,4,6,12), (3,6,12), (2,3,6,12), (4,3,6,12), (2,4,3,6,12).


MATHEMATICA

a[n_] := Sum[ MoebiusMu[n/d] * 2^(DivisorSigma[0, d]  1), {d, Divisors[n]}]; Table[a[n], {n, 1, 92}](* JeanFrançois Alcover, Nov 30 2011, after Vladeta Jovovic *)


PROG

(Haskell)  following Vladeta Jovovic's formula.
a069626 n = sum $
map (\d > (a008683 (n `div` d)) * 2 ^ (a000005 d  1)) $ a027750_row n
 Reinhard Zumkeller, Jun 12 2015, Feb 07 2011


CROSSREFS

Sequence in context: A055205 A161686 A289621 * A249274 A205443 A069359
Adjacent sequences: A069623 A069624 A069625 * A069627 A069628 A069629


KEYWORD

nonn,nice,easy


AUTHOR

Amarnath Murthy, Mar 27 2002


EXTENSIONS

Corrected and extended by Naohiro Nomoto, Apr 25 2002


STATUS

approved



