A349281 -id:A349281 - cp's OEIS frontend

A349282 a(n) is the least k such that A349281(k) = n.

1, 2, 4, 12, 36, 144, 576, 2880, 14400, 100800, 705600, 6350400, 45158400, 406425600, 3657830400, 40236134400, 442597478400, 5753767219200, 74798973849600, 1271582555443200, 21616903442534400, 410721165408153600, 7803702142754918400, 179485149283363123200, 4128158433517351833600

Offset: 0

Amiram Eldar, Nov 13 2021

nonn

			a(2) = 4 since A349281(4) = 2 and A349281(k) != 2 for all k < 4.

Amiram Eldar, Table of n, a(n) for n = 0..40

Cf. A349260, A349281.

Subsequence of A025487.

f[p_, e_] := DivisorSigma[0, e]; d[1] = 0; d[n_] := Plus @@ f @@@ FactorInteger[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], k = 0, n = 1, i}, While[k < len && n < nmax, i = d[n] + 1; If[i <= len && s[[i]] == 0, k++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[11, 10^6]

A381401 a(n) is the number of (possibly non-distinct) prime elements in the multiset of bases and exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Feb 24 2025

			a(144) = 3 because the prime factorization of 144 is 2^4*3^2 and the multiset of these bases and exponents is {2, 2, 3, 4}, containing 3 primes.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A106490, A349281, A381398, A381399.

A381401[n_] := Count[FactorInteger[n], _?PrimeQ, {2}];
Array[A381401, 100]

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 15 2021

The total number of prime powers (not including 1) that divide n is A001222(n).

The least number k such that a(k) = m is A122756(m).

			12 has 4 bi-unitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.

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

Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.
Similar sequences: A001222, A349258, A349281.
Cf. A083342, A179119.

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

a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 2023

Additive with a(p^e) = e if e is odd, and e-1 if e is even.
a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
a(n) = A001222(n) - A162641(n). - Amiram Eldar, May 18 2023
From Amiram Eldar, Sep 29 2023: (Start)
a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)

cp's OEIS Frontend

A349282 a(n) is the least k such that A349281(k) = n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A381401 a(n) is the number of (possibly non-distinct) prime elements in the multiset of bases and exponents in the prime factorization of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula