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

1,2

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)

Links

Table of n, a(n) for n=1..57.

Formula

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

Example

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

Prog

(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

Crossrefs

Cf. A000203, A004125, A024916, A111490.

Sequence in context: A088937 A295362 A162383 * A024679 A187973 A190488

Adjacent sequences: A123324 A123325 A123326 * A123328 A123329 A123330

Keyword

easy,nonn

Author

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

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.

Status

approved