A383959 -id:A383959 - cp's OEIS frontend

A383960 The number of prime powers p^e having the property that e is an infinitary divisor of the p-adic valuation of n.

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, May 16 2025

First differs from A238949 at n = 64.

First differs from A383959 at n = 256.

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

Cf. A037445, A085548, A238949, A383760, A383865, A383959.

f[p_, e_] := 2^DigitCount[e, 2, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 0; ff[p_, e_] := d[e]; a[n_] := Plus @@ ff @@@ FactorInteger[n]; Array[a, 100]

d(n) = vecprod(apply(x -> 1 << hammingweight(x), factor(n)[, 2]));
a(n) = vecsum(apply(x -> d(x), factor(n)[, 2]));

Additive with a(p^e) = A037445(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)*A037445(e)/p^e = 0.92752481299257205938... .

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... .

cp's OEIS Frontend

A383960 The number of prime powers p^e having the property that e is an infinitary divisor of the p-adic valuation of n.

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