A125029 -id:A125029 - 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

A125030 a(n) = sum of exponents in the prime factorization of n that are noncomposite.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 1, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 1, 0, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3

Offset: 1

Leroy Quet, Nov 16 2006

			a(720) = 3, since the prime factorization of 720 is 2^4 * 3^2 * 5^1 and two of the exponents in this factorization are noncomposites (the exponents 2 and 1, whose sum is 3).

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

Cf. A077761, A001222, A125029, A125071.

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

A125030(n) = vecsum(apply(e -> if((1==e)||isprime(e),e,0), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017

From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = e if e is composite, and 0 otherwise.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = - P(2) + Sum_{p prime} p * (P(p) - P(p+1)) = 0.52262278983683613884..., where P(s) is the prime zeta function. (End)

Extended by Ray Chandler, Nov 19 2006

A125072 a(n) = number of exponents in the prime-factorization of n which are triangular numbers.

Original entry on oeis.org

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, 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, 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

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) = 2.

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, A125073, A125029, A125070.

f[n_] := Length @ 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.
A125072(n) = vecsum(apply(e -> A010054(e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 08 2017

Additive with a(p^e) = A010054(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} (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

Extended by Ray Chandler, Nov 19 2006

A384421 The number of exponentially squarefree prime powers (not including 1) that unitarily divide n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 1, 0, 2, 1, 3, 2, 2, 2

Offset: 1

Amiram Eldar, May 28 2025

First differs from A125029 at n = 64.
A number k unitarily divides n if k|n and gcd(k, n/k) = 1.
The number of unitary divisors of n that are larger than 1 and are terms in A384419.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008966, A125029, A383959, A384419, A384420.

f[p_, e_] := If[SquareFreeQ[e], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(issquarefree, factor(n)[, 2]));

Additive with a(p^e) = A008966(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A008966(e)/p^e = 0.40780808646244052181... .

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

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A125030 a(n) = sum of exponents in the prime factorization of n that are noncomposite.

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A125072 a(n) = number of exponents in the prime-factorization of n which are triangular numbers.

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A384421 The number of exponentially squarefree prime powers (not including 1) that unitarily divide n.

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