A375931 The product of the prime powers in the prime factorization of n that have an exponent that is equal to the maximum exponent in this factorization.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 13, 14, 15, 16, 17, 9, 19, 4, 21, 22, 23, 8, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 8, 41, 42, 43, 4, 9, 46, 47, 16, 49, 25, 51, 4, 53, 27, 55, 8, 57, 58, 59, 4, 61, 62, 9, 64, 65, 66, 67, 4, 69, 70, 71

Offset

1,2

Comments

Differs from A327526 at n = 12, 20, 28, 40, 44, 45, ... .

Each positive number appears in this sequence either once or infinitely many times:

1. If m is squarefree then the only solution to a(x) = m is x = m.

2. If m = s^k is a power of a squarefree number s with k >= 2, then x = m * i is a solution to a(x) = m for all numbers i that are k-free numbers (i.e., having exponents in their prime factorizations that are all less than k) that are coprime to m.

Links

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

Formula

If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i = max_{j} e_j]), where [] is the Iverson bracket.

a(n) = A261969(n)^A051903(n).

a(n) = n / A375932(n).

a(n) = n if and only if n is a power of a squarefree number (A072774).

A051903(a(n)) = A051903(n).

omega(a(n)) = A362611(n).

omega(a(n)) = 1 if and only if n is in A356862.

Omega(a(n)) = A051903(n) * A362611(n).

a(n!) = A060818(n) for n != 3.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/Pi^2 = 0.303963... (A104141).

Example

180 = 2^2 * 3^2 * 5, and the maximum exponent in the prime factorization of 180 is 2, which is the exponent of its prime factors 2 and 3. Therefore a(180) = 2^2 * 3^2 = (2*3)^2 = 36.

Mathematica

a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; (Times @@ p[[i]])^m]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] == m, p[i], 1))^m); }

Crossrefs

Cf. A001221 (omega), A001222 (Omega), A005117, A051903, A060818, A072774, A104141, A261969, A327526, A356862, A362611, A375932.

Sequence in context: A043271 A333921 A278063 * A373231 A334684 A062759

Adjacent sequences: A375928 A375929 A375930 * A375932 A375933 A375934

Keyword

nonn,easy

Author

Amiram Eldar, Sep 03 2024

Status

approved