A123327 a(n) = A000203(n) + A004125(n).

1, 3, 5, 8, 10, 15, 16, 23, 25, 31, 34, 45, 42, 55, 60, 67, 69, 86, 84, 103, 102, 113, 122, 145, 134, 154, 165, 180, 181, 210, 199, 230, 232, 251, 266, 289, 271, 308, 325, 348, 339, 380, 369, 412, 417, 430, 451, 498, 471, 513, 521, 552, 559, 612, 601, 640, 633

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Sep 26 2006; Juri-Stepan Gerasimov, Jul 02 2009

Keywords

• easy
• nonn

Comments

Another definition for this sequence: Let M be the matrix defined in A111490. Sequence gives M(1,1), M(1,2) + M(2,2), M(1,3) + M(2,3) + M(3,3), etc., i.e. a(n)= Sum_{i=1..n} M(i,n).

Proof from Hartmut F. W. Hoft, Feb 02 2014 that the two definitions agree: (Start)

For all n>=1 the following simplifications hold for the partial sums of the two sequences:

sum[1..n] a(k) = sum[1..n] A000203(k) + sum[1..n] A004125(k)

= A024916(n) + sum[1..n] A004125(k)

= n^2 + sum[1..n-1] A004125(k)

= sum[1..n] A123327(k).

An inductive argument then shows that the two definitions agree.

(End)

Examples

			1(=1+0), 3(=3+0), 5(=4+1), 8(=7+1), 10(=6+4), 15(=12+3), 16(=8+8), etc.
		

Crossrefs

Cf. A000203, A004125, A024916, A111490.

Programs

Mathematica

Lim=57;s2=Table[Sum[Mod[n, k], {k, 2, n-1}], {n, Lim}];Table[DivisorSigma[1, n]+s2[[n]],{n,Lim}] (* James C. McMahon, Nov 20 2024 *)

Python

from math import isqrt
def A123327(n): return n**2+((s:=isqrt(n-1))**2*(s+1)-sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

Formula

a(n) = A000290(n) - A024916(n-1), n > 1. - Omar E. Pol, Jan 29 2014

Extensions

Corrected (83 replaced by 103) by R. J. Mathar, May 21 2010

Edited by N. J. A. Sloane, Feb 02 2014, merging A162383 from Juri-Stepan Gerasimov with the present sequence. Thanks to Omar E. Pol for noticing the duplication.