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:
= n^2 + sum[1..n-1] A004125(k)
= sum[1..n] A123327(k).
An inductive argument then shows that the two definitions agree.
(End)
FORMULA
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
KEYWORD
easy,nonn
AUTHOR
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