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
Examples
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.
Programs
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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
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Mathematica
Table[DivisorSigma[1, n] - DivisorSigma[0, n] + n, {n, 100}] (* Vincenzo Librandi, Sep 21 2015 *) -
PARI
a(n) = sigma(n) - numdiv(n) + n; \\ Michel Marcus, Oct 29 2022
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
Extensions
Corrected and extended by R. J. Mathar, Apr 09 2008
Edited by Jon E. Schoenfield, Sep 21 2015
Comments