A367859 Multiset multiplicity cokernel (MMC) of n. Product of (greatest prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

1, 2, 3, 2, 5, 9, 7, 2, 3, 25, 11, 6, 13, 49, 25, 2, 17, 6, 19, 10, 49, 121, 23, 6, 5, 169, 3, 14, 29, 125, 31, 2, 121, 289, 49, 9, 37, 361, 169, 10, 41, 343, 43, 22, 15, 529, 47, 6, 7, 10, 289, 26, 53, 6, 121, 14, 361, 841, 59, 50, 61, 961, 21, 2, 169, 1331

Offset: 1

Author

Gus Wiseman, Dec 03 2023

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define the multiset multiplicity cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.

Examples

			90 has prime factorization 2^1*3^2*5^1, so for k = 1 we have 5^2, and for k = 2 we have 3^1, so a(90) = 75.
		

Crossrefs

Positions of 2's are A000079 without 1.

Positions of 3's are A000244 without 1.

Positions of primes (including 1) are A000961.

Depends only on rootless base A052410, see A007916.

Positions of prime powers are A072774.

Positions of squarefree numbers are A130091.

For kernel instead of cokernel we have A367580, ranks of A367579.

Rows of A367858 have this rank, sum A367860, max A061395, min A367587.

A007947 gives squarefree kernel.

A027746 lists prime factors, length A001222, indices A112798.

A027748 lists distinct prime factors, length A001221, indices A304038.

A071625 counts distinct prime exponents.

A124010 gives multiset of multiplicities (prime signature), sorted A118914.

Cf. A005117, A020639, A052409, A175781, A181819, A353742, A367584, A367585.

Programs

Mathematica

mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,30}]

Formula

a(n^k) = a(n) for all positive integers n and k.

If n is squarefree, a(n) = A006530(n)^A001222(n).

A055396(a(n)) = A367587(n).

A056239(a(n)) = A367860(n).

A061395(a(n)) = A061395(n).

A001222(a(n)) = A001221(n).

A001221(a(n)) = A071625(n).

A071625(a(n)) = A323022(n).