A367683 Numbers whose sorted prime signature is the same as the multiset multiplicity kernel of their prime indices.

1, 2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 38, 40, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 102, 106, 110, 112, 114, 118, 122, 125, 126, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 225

Offset: 1

Author

Gus Wiseman, Nov 30 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:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   34: {1,7}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   46: {1,9}
   58: {1,10}
   62: {1,11}
   66: {1,2,5}
   70: {1,3,4}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, 20 X 20 color coded list of terms, color function: gray = 1, red = prime, gold = composite prime power, green = squarefree composite, blue-purple = numbers neither squarefree nor prime powers. Bright green = primorial, light green = even squarefree semiprime, light blue = highly composite, middle blue = in A055932, purple = squareful but not a prime power.
• Michael De Vlieger, 485 X 485 = 235225-term raster with the same color code as above.

Crossrefs

Squarefree terms are A039956.

The LHS is A118914, unsorted A124010.

Prime-power terms are A307539.

The RHS is A367579, ranks A367580, sum A367581, max A367583.

Partitions of this type are counted by A367682.

A007947 gives squarefree kernel.

A112798 lists prime indices, length A001222, sum A056239, reverse A296150.

A181819 gives prime shadow, with an inverse A181821.

A238747 gives prime metasignature, reversed A353742.

A304038 lists distinct prime indices, length A001221, sum A066328.

A367582 counts partitions by sum of multiset multiplicity kernel.

Cf. A000720, A000961, A005117, A051904, A052409, A071625, A072774, A130091, A367584, A367586, A367587.

Programs

Mathematica

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Select[Range[100], #==1||Sort[Last/@FactorInteger[#]] == mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]]&]