A068312 Arithmetic derivative of triangular numbers.

0, 0, 1, 5, 7, 8, 10, 32, 60, 39, 16, 61, 71, 20, 71, 244, 212, 111, 123, 143, 247, 131, 34, 380, 520, 155, 378, 621, 275, 247, 263, 1008, 1280, 271, 239, 951, 795, 56, 343, 1256, 1004, 431, 451, 581, 1443, 942, 70, 2092, 2492, 840

Offset

0,4

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Formula

a(n) = A003415(A000217(n)).

For n > 1: a(n) = A258197(n,2) = A258197(n,n-2). - Reinhard Zumkeller, May 26 2015

Example

a(7) = d(7*8/2) = d(28) = d(2*14) = d(2)*14 + 2*d(14) = 1*14 + 2*d(2*7) = 14 + 2*(2*d(7) + d(2)*7) = 14 + 2*(2*1 + 1*7) = 14 + 2*9 = 14 + 18 = 32, where d(n) = A003415(n) with d(1) = 0, d(prime) = 1 and d(m*n) = d(m)*n + m*d(n).

Mathematica

a[0] = a[1] = 0; a[n_] := (n*(n+1)/2) * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n*(n+1)/2]); Array[a, 100, 0] (* Amiram Eldar, May 14 2025 *)

Prog

(Haskell)
a068312 = a003415 . a000217  -- Reinhard Zumkeller, May 26 2015
(Python)
from sympy import factorint
def A068312(n): return 0 if n <= 1 else ((n+1)*sum((n*e//p for p, e in factorint(n).items()))+ sum(((n+1)*e//p for p, e in factorint(n+1).items()))*n - (n*(n+1)//2))//2 # Chai Wah Wu, Jun 24 2022

Crossrefs

Cf. A000217, A003415, A007318, A258197.

Sequence in context: A314373 A192270 A056657 * A082097 A088299 A287068

Adjacent sequences: A068309 A068310 A068311 * A068313 A068314 A068315

Keyword

nonn,easy

Author

Reinhard Zumkeller, Feb 25 2002

Extensions

a(0)=0 prepended by Reinhard Zumkeller, May 26 2015

Status

approved