A367580 Multiset multiplicity kernel (MMK) of n. Product of (least 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, 4, 7, 2, 3, 4, 11, 6, 13, 4, 9, 2, 17, 6, 19, 10, 9, 4, 23, 6, 5, 4, 3, 14, 29, 8, 31, 2, 9, 4, 25, 4, 37, 4, 9, 10, 41, 8, 43, 22, 15, 4, 47, 6, 7, 10, 9, 26, 53, 6, 25, 14, 9, 4, 59, 18, 61, 4, 21, 2, 25, 8, 67, 34, 9, 8, 71, 6, 73, 4, 15, 38

Offset: 1

Author

Gus Wiseman, Nov 26 2023

Keywords

• nonn

Comments

As an operation on multisets, this is represented by A367579.

Examples

			90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so a(90) = 12.
		

Crossrefs

Positions of 2's are A000079 without 1.

Positions of 3's are A000244 without 1.

Positions of primes (including 1) are A000961.

Positions of prime(k) are prime powers prime(k)^i, rows of A051128.

Depends only on rootless base A052410, see A007916.

Positions of prime powers are A072774.

Positions of squarefree numbers are A130091.

Agrees with A181819 at positions A367683, counted by A367682.

Rows of A367579 have this rank, sum A367581, max A367583, min A055396.

Positions of first appearances are A367584, sorted A367585.

Positions of powers of 2 are A367586.

Divides n at positions A367685, counted by A367684.

The opposite version (cokernel) is A367859.

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. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.

Programs

Mathematica

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,100}]

Formula

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

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

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

If n is squarefree, a(n) = A020639(n)^A001222(n).

A056239(a(n)) = A367581(n).