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A125073
a(n) = sum of the exponents in the prime factorization of n which are triangular numbers.
4
0, 1, 1, 0, 1, 2, 1, 3, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 4, 0, 2, 3, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 4, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 1, 6, 2, 3, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 4, 3
OFFSET
1,6
FORMULA
Additive with a(p^e) = A010054(e)*e. - Antti Karttunen, Jul 08 2017
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=2} (k*(k+1)/2) * (P(k*(k+1)/2) - P(k*(k+1)/2 + 1)) = -0.10099019472003733178..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 28 2023
EXAMPLE
The prime factorization of 360 is 2^3 *3^2 *5^1. There are two exponents in this factorization which are triangular numbers, 1 and 3. So a(360) = 1 + 3 = 4.
MATHEMATICA
f[n_] := Plus @@ Select[Last /@ FactorInteger[n], IntegerQ[Sqrt[8# + 1]] &]; Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)
PROG
(PARI)
A010054(n) = issquare(8*n + 1); \\ This function from Michael Somos, Apr 27 2000.
A125073(n) = vecsum(apply(e -> (A010054(e)*e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 08 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Leroy Quet, Nov 18 2006
EXTENSIONS
Extended by Ray Chandler, Nov 19 2006
STATUS
approved