A133699 -id:A133699 - cp's OEIS frontend

A133698 Triangle, diagonal = A001227 with the rest zeros.

1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gary W. Adamson, Sep 21 2007

Lower triangular part of an infinite matrix with A001227 (number of odd divisors of n) as the main diagonal, and the rest filled with zeros. - Redacted from the original formula given by the author. - Antti Karttunen, Jan 18 2025

			First few rows of the triangle are:
  1;
  0, 1;
  0, 0, 2
  0, 0, 0, 1;
  0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 3;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2;
  ...

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 rows of the triangle

Cf. A001227, A130209, A133699.

A000265(n) = (n>>valuation(n,2));
A133698(n) = if(ispolygonal(n,3), numdiv(A000265((sqrtint(1+(n*8))-1)/2)), 0); \\ Antti Karttunen, Jan 18 2025

Offset corrected from 0 to 1 and data section extended to a(105) by Antti Karttunen, Jan 18 2025

A133700 A051731 * A001227; a(n) = Sum_{d|n} A001227(d).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Sep 21 2007

			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, ...).

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

Cf. A000005, A001227, A051731, A133699.

Cf. A001620, A082633.

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 *)

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

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

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

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

cp's OEIS Frontend

A133698 Triangle, diagonal = A001227 with the rest zeros.

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