A137319 Start with the set of natural numbers. Add 1 to every 2nd term, 2 to every 3rd term, 3 to every 4th term, etc.

1, 3, 5, 8, 9, 14, 13, 19, 19, 24, 21, 34, 25, 34, 35, 42, 33, 51, 37, 56, 49, 54, 45, 76, 53, 64, 63, 78, 57, 94, 61, 89, 77, 84, 79, 118, 73, 94, 91, 122, 81, 130, 85, 122, 117, 114, 93, 162, 103, 137, 119, 144, 105, 166, 123, 168, 133, 144, 117, 216, 121, 154, 161, 184

Offset

1,2

Comments

The generating function is the sum of the generating functions in A000027 and A065608. - R. J. Mathar, Apr 09 2008

Links

Table of n, a(n) for n=1..64.

Formula

a(n) = n + A065608(n). - R. J. Mathar, Apr 09 2008

a(n) = Sum_{k=1..n} k^(1-ceiling(n/k)+floor(n/k)). - Wesley Ivan Hurt, May 24 2021

Example

Start with the natural numbers:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
add 1 to every 2nd term:
   1,  3,  3,  5,  5,  7,  7,  9,  9, 11, ...
add 2 to every 3rd term:
   1,  3,  5,  5,  5,  9,  7,  9, 11, 11, ...
add 3 to every 4th term:
   1,  3,  5,  8,  5,  9,  7, 12, 11, 11, ...
add 4 to every 5th term:
   1,  3,  5,  8,  9,  9,  7, 12, 11, 15, ...
etc.

Maple

A137319 := proc(n) local a, k ; a := n ; for k from 2 to n do if n mod k = 0 then a := a+k-1 ; fi ; od: a; end: seq(A137319(n), n=1..100) ; # R. J. Mathar, Apr 09 2008

Mathematica

Table[DivisorSigma[1, n] - DivisorSigma[0, n] + n, {n, 100}] (* Vincenzo Librandi, Sep 21 2015 *)

Prog

(PARI) a(n) = sigma(n) - numdiv(n) + n; \\ Michel Marcus, Oct 29 2022

Crossrefs

Cf. A000027, A065608.

Sequence in context: A186621 A002159 A050094 * A138808 A185456 A308405

Adjacent sequences: A137316 A137317 A137318 * A137320 A137321 A137322

Keyword

easy,nonn

Author

Ctibor O. Zizka, Apr 06 2008

Extensions

Corrected and extended by R. J. Mathar, Apr 09 2008

Edited by Jon E. Schoenfield, Sep 21 2015

Status

approved