A367586 Numbers whose prime indices have a multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) that is all ones {1,1,...}. Positions of powers of 2 in A367580.

1, 2, 4, 6, 8, 10, 14, 16, 22, 26, 30, 32, 34, 36, 38, 42, 46, 58, 62, 64, 66, 70, 74, 78, 82, 86, 94, 100, 102, 106, 110, 114, 118, 122, 128, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 196, 202, 206, 210, 214, 216, 218, 222

Offset: 1

Author

Gus Wiseman, Nov 30 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 kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

Examples

			We have MMK({1,1,2,2}) = {1,1} so 36 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   14: {1,4}
   16: {1,1,1,1}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   38: {1,8}
   42: {1,2,4}
		

Crossrefs

Contains all prime powers A000961 and squarefree numbers A005117.

Partitions of this type (uniform containing 1) are counted by A097986.

Positions of all one rows {1,1,...} in A367579.

Positions of powers of 2 in A367580.

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 prime signature, sorted A118914.

A367581 gives multiset multiplicity kernel sum, max A367583, min A055396.

Cf. A001597, A051904, A072774, A130091, A175781, A367584, A367587, A367685.

Programs

Maple

isA := proc(n) z := padic:-ordp(n, 2); andseq(z=p[2], p in ifactors(n)[2]) end:
select(isA, [seq(1..222)]);  # Peter Luschny, Jun 10 2025

Mathematica

Select[Range[100], #==1||EvenQ[#]&&SameQ@@Last/@FactorInteger[#]&]

Formula

Consists of 1 and all even terms of A072774 (powers of squarefree numbers).