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A068312
Arithmetic derivative of triangular numbers.
4
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
COMMENTS
For n > 1: a(n) = A258197(n,2) = A258197(n,n-2). - Reinhard Zumkeller, May 26 2015
LINKS
FORMULA
a(n) = A003415(A000217(n)).
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).
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 25 2002
EXTENSIONS
a(0)=0 prepended by Reinhard Zumkeller, May 26 2015
STATUS
approved

  NODES
orte 1
see 1
Story 1