%I #24 Sep 28 2023 04:23:50

%S 0,1,1,0,1,2,1,1,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,2,0,2,1,1,1,3,1,0,2,2,

%T 2,0,1,2,2,2,1,3,1,1,1,2,1,1,0,1,2,1,1,2,2,2,2,2,1,2,1,2,1,1,2,3,1,1,

%U 2,3,1,1,1,2,1,1,2,3,1,1,0,2,1,2,2,2,2,2,1,2,2,1,2,2,2,1,1,1,1,0,1,3,1,2,3

%N a(n) = number of exponents in the prime-factorization of n which are triangular numbers.

%H Antti Karttunen, <a href="/A125072/b125072.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F Additive with a(p^e) = A010054(e). - _Antti Karttunen_, Jul 08 2017

%F 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} (P(k*(k+1)/2) - P(k*(k+1)/2 + 1)) = -0.34517646457715166126..., where P(s) is the prime zeta function. - _Amiram Eldar_, Sep 28 2023

%e 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) = 2.

%t f[n_] := Length @ Select[Last /@ FactorInteger[n], IntegerQ[Sqrt[8# + 1]] &];Table[f[n], {n, 110}] (* _Ray Chandler_, Nov 19 2006 *)

%o (PARI)

%o A010054(n) = issquare(8*n + 1); \\ This function from _Michael Somos_, Apr 27 2000.

%o A125072(n) = vecsum(apply(e -> A010054(e), factorint(n)[, 2])); \\ _Antti Karttunen_, Jul 08 2017

%Y Cf. A010054, A077761, A125073, A125029, A125070.

%K nonn,easy

%O 1,6

%A _Leroy Quet_, Nov 18 2006

%E Extended by _Ray Chandler_, Nov 19 2006