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

Author

Reinhard Zumkeller, Feb 25 2002

Keywords

• nonn
• easy

Examples

			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).
		

Links

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

Crossrefs

Cf. A000217, A003415, A007318, A258197.

Programs

Haskell

a068312 = a003415 . a000217  -- Reinhard Zumkeller, May 26 2015

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 *)

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

Formula

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

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

Extensions

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