A125072 -id:A125072 - cp's OEIS frontend

A101436 Number of exponents in prime factorization of n which are primes.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0

Offset: 1

Leroy Quet, Jan 18 2005

First occurrence of k: 1,4,36,900,44100 (A061742). - Robert G. Wilson v, Jan 25 2005

			36 = 2^2 *3^2. Since 2 is a prime and occurs twice as an exponent in the prime factorization of 36, a(36) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A010051, A061742, A123391, A125029, A125070, A125072.

f[n_] := Length[ Select[ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]], PrimeQ[ # ] &]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Jan 25 2005 *)
Table[Count[Transpose[FactorInteger[n]][[2]],?PrimeQ],{n,120}] (* _Harvey P. Dale, Mar 21 2016 *)

A101436(n) = vecsum(apply(e -> isprime(e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 19 2017

Additive with a(p^e) = A010051(e). - Antti Karttunen, Jul 19 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (P(p)-P(p+1)) = 0.39847584805803104040..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 29 2023

More terms from Robert G. Wilson v, Jan 25 2005

A125073 a(n) = sum of the exponents in the prime factorization of n which are triangular numbers.

Original entry on oeis.org

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

Leroy Quet, Nov 18 2006

			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.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A010054, A077761, A125072, A125030, A125071.

f[n_] := Plus @@ Select[Last /@ FactorInteger[n], IntegerQ[Sqrt[8# + 1]] &];Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)

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

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

Extended by Ray Chandler, Nov 19 2006

cp's OEIS Frontend

A101436 Number of exponents in prime factorization of n which are primes.

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