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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.
10
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
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
Sequence in context: A186621 A002159 A050094 * A138808 A185456 A308405
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