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

%I A367683 #12 Dec 01 2023 15:57:03
%S A367683 1,2,6,9,10,12,14,18,22,26,30,34,38,40,42,46,58,62,66,70,74,78,82,86,
%T A367683 90,94,102,106,110,112,114,118,122,125,126,130,134,138,142,146,154,
%U A367683 158,166,170,174,178,182,186,190,194,198,202,206,210,214,218,222,225
%N A367683 Numbers whose sorted prime signature is the same as the multiset multiplicity kernel of their prime indices.
%C A367683 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.
%H A367683 Michael De Vlieger, <a href="/A367683/b367683.txt">Table of n, a(n) for n = 1..10000</a>
%H A367683 Michael De Vlieger, <a href="/A367683/a367683.png">20 X 20 color coded list of terms</a>, 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.
%H A367683 Michael De Vlieger, <a href="/A367683/a367683_1.png">485 X 485 = 235225-term raster</a> with the same color code as above.
%e A367683 The terms together with their prime indices begin:
%e A367683     1: {}
%e A367683     2: {1}
%e A367683     6: {1,2}
%e A367683     9: {2,2}
%e A367683    10: {1,3}
%e A367683    12: {1,1,2}
%e A367683    14: {1,4}
%e A367683    18: {1,2,2}
%e A367683    22: {1,5}
%e A367683    26: {1,6}
%e A367683    30: {1,2,3}
%e A367683    34: {1,7}
%e A367683    38: {1,8}
%e A367683    40: {1,1,1,3}
%e A367683    42: {1,2,4}
%e A367683    46: {1,9}
%e A367683    58: {1,10}
%e A367683    62: {1,11}
%e A367683    66: {1,2,5}
%e A367683    70: {1,3,4}
%t A367683 mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367683 Select[Range[100], #==1||Sort[Last/@FactorInteger[#]] == mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]]&]
%Y A367683 Squarefree terms are A039956.
%Y A367683 The LHS is A118914, unsorted A124010.
%Y A367683 Prime-power terms are A307539.
%Y A367683 The RHS is A367579, ranks A367580, sum A367581, max A367583.
%Y A367683 Partitions of this type are counted by A367682.
%Y A367683 A007947 gives squarefree kernel.
%Y A367683 A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A367683 A181819 gives prime shadow, with an inverse A181821.
%Y A367683 A238747 gives prime metasignature, reversed A353742.
%Y A367683 A304038 lists distinct prime indices, length A001221, sum A066328.
%Y A367683 A367582 counts partitions by sum of multiset multiplicity kernel.
%Y A367683 Cf. A000720, A000961, A005117, A051904, A052409, A071625, A072774, A130091, A367584, A367586, A367587.
%K A367683 nonn
%O A367683 1,2
%A A367683 _Gus Wiseman_, Nov 30 2023