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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.

%I A137319 #16 Oct 29 2022 16:06:54
%S A137319 1,3,5,8,9,14,13,19,19,24,21,34,25,34,35,42,33,51,37,56,49,54,45,76,
%T A137319 53,64,63,78,57,94,61,89,77,84,79,118,73,94,91,122,81,130,85,122,117,
%U A137319 114,93,162,103,137,119,144,105,166,123,168,133,144,117,216,121,154,161,184
%N 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.
%C A137319 The generating function is the sum of the generating functions in A000027 and A065608. - _R. J. Mathar_, Apr 09 2008
%F A137319 a(n) = n + A065608(n). - _R. J. Mathar_, Apr 09 2008
%F A137319 a(n) = Sum_{k=1..n} k^(1-ceiling(n/k)+floor(n/k)). - _Wesley Ivan Hurt_, May 24 2021
%e A137319 Start with the natural numbers:
%e A137319    1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
%e A137319 add 1 to every 2nd term:
%e A137319    1,  3,  3,  5,  5,  7,  7,  9,  9, 11, ...
%e A137319 add 2 to every 3rd term:
%e A137319    1,  3,  5,  5,  5,  9,  7,  9, 11, 11, ...
%e A137319 add 3 to every 4th term:
%e A137319    1,  3,  5,  8,  5,  9,  7, 12, 11, 11, ...
%e A137319 add 4 to every 5th term:
%e A137319    1,  3,  5,  8,  9,  9,  7, 12, 11, 15, ...
%e A137319 etc.
%p A137319 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
%t A137319 Table[DivisorSigma[1, n] - DivisorSigma[0, n] + n, {n, 100}] (* _Vincenzo Librandi_, Sep 21 2015 *)
%o A137319 (PARI) a(n) = sigma(n) - numdiv(n) + n; \\ _Michel Marcus_, Oct 29 2022
%Y A137319 Cf. A000027, A065608.
%K A137319 easy,nonn
%O A137319 1,2
%A A137319 _Ctibor O. Zizka_, Apr 06 2008
%E A137319 Corrected and extended by _R. J. Mathar_, Apr 09 2008
%E A137319 Edited by _Jon E. Schoenfield_, Sep 21 2015