A367768 Numbers k such that MMK(k) = MMK(i) for some i < k, where MMK is multiset multiplicity kernel A367580.

4, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 27, 32, 33, 34, 36, 38, 39, 40, 42, 46, 48, 49, 50, 51, 54, 55, 56, 57, 58, 62, 64, 65, 66, 69, 70, 72, 74, 75, 78, 80, 81, 82, 84, 85, 86, 87, 88, 91, 93, 94, 95, 96, 98, 100, 102, 104, 106, 108, 110, 111, 112, 114

Offset: 1

Author

Gus Wiseman, Dec 01 2023

Keywords

• nonn

Comments

We define the multiset multiplicity kernel (MMK) of a positive integer n to be the 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. For example, 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 MMK(90) = 12. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

Examples

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   32: {1,1,1,1,1}
   33: {2,5}
   34: {1,7}
   36: {1,1,2,2}
		

Crossrefs

The squarefree case is A073486, complement A073485.

The MMK triangle is A367579, sum A367581, min A055396, max A367583.

Sorted positions of non-first appearances in A367580.

The complement is A367585, sorted version of A367584.

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.

Cf. A051904, A072774, A130091, A181819, A246547, A367685.

Programs

Mathematica

nn=100;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
qq=Table[Times@@mmk[Join @@ ConstantArray@@@FactorInteger[n]],{n,nn}];
Select[Range[nn], MemberQ[Take[qq,#-1], qq[[#]]]&]

Formula

A367580(a(k)) = A367580(i) for some i < a(k).