A133700 - OEIS

A133700

A051731 * A001227; a(n) = Sum_{d|n} A001227(d).

1, 2, 3, 3, 3, 6, 3, 4, 6, 6, 3, 9, 3, 6, 9, 5, 3, 12, 3, 9, 9, 6, 3, 12, 6, 6, 10, 9, 3, 18, 3, 6, 9, 6, 9, 18, 3, 6, 9, 12, 3, 18, 3, 9, 18, 6, 3, 15, 6, 12, 9, 9, 3, 20, 9, 12, 9, 6, 3, 27, 3, 6, 18, 7, 9, 18, 3, 9, 9, 18, 3, 24, 3, 6, 18, 9, 9, 18, 3, 15, 15, 6, 3, 27, 9, 6, 9, 12, 3, 36, 9, 9, 9

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OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

Inverse Möbius transform of A001227, the number of odd divisors of n. Row sums of triangle A133699.

Dirichlet g.f. (zeta(s))^3*(1-1/2^s). - R. J. Mathar, Feb 07 2011

a(n) = Sum_{d|n} A001227(d). - Antti Karttunen, Sep 27 2018

Sum_{k=1..n} a(k) ~ n/4 * (log(n)^2 + (6*g - 2 + 2*log(2))*log(n) + 2 + 6*g^2 - log(2)^2 - 2*log(2) + 6*g*(log(2) - 1) - 6*sg1), where g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Feb 02 2019

G.f.: Sum_{k>=1} tau(k)*x^k/(1 - x^(2*k)), where tau = A000005. - Ilya Gutkovskiy, Sep 13 2019

Multiplicative with a(2^e) = e+1, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Oct 28 2023

EXAMPLE

a(4) = sum of row 4 terms of triangle A133699: (1 + 1 + 0 + 1) = (1, 1, 0, 1) dot (1, 1, 2, 1), where A001227 = (1, 1, 2, 1, 2, 2, 2, 1, 3, ...).

MATHEMATICA

f[p_, e_] := (e+1)*(e+2)/2; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 28 2023 *)

PROG

(PARI) A133700(n) = sumdiv(n, d, numdiv(d>>valuation(d, 2))); \\ Antti Karttunen, Sep 27 2018

CROSSREFS

Cf. A000005, A001227, A051731, A133699.

Cf. A001620, A082633.

Sequence in context: A373093 A134187 A078644 * A087688 A126854 A115206

Adjacent sequences: A133697 A133698 A133699 * A133701 A133702 A133703

KEYWORD

nonn,easy,mult

AUTHOR

Gary W. Adamson, Sep 21 2007

EXTENSIONS

More terms from R. J. Mathar, Jan 19 2009

Second, equivalent formula added to the definition by Antti Karttunen, Sep 27 2018

STATUS

approved