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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.

%I A367586 #12 Jun 10 2025 16:32:55
%S A367586 1,2,4,6,8,10,14,16,22,26,30,32,34,36,38,42,46,58,62,64,66,70,74,78,
%T A367586 82,86,94,100,102,106,110,114,118,122,128,130,134,138,142,146,154,158,
%U A367586 166,170,174,178,182,186,190,194,196,202,206,210,214,216,218,222
%N 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.
%C A367586 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.
%C A367586 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.
%F A367586 Consists of 1 and all even terms of A072774 (powers of squarefree numbers).
%e A367586 We have MMK({1,1,2,2}) = {1,1} so 36 is in the sequence.
%e A367586 The terms together with their prime indices begin:
%e A367586     1: {}
%e A367586     2: {1}
%e A367586     4: {1,1}
%e A367586     6: {1,2}
%e A367586     8: {1,1,1}
%e A367586    10: {1,3}
%e A367586    14: {1,4}
%e A367586    16: {1,1,1,1}
%e A367586    22: {1,5}
%e A367586    26: {1,6}
%e A367586    30: {1,2,3}
%e A367586    32: {1,1,1,1,1}
%e A367586    34: {1,7}
%e A367586    36: {1,1,2,2}
%e A367586    38: {1,8}
%e A367586    42: {1,2,4}
%p A367586 isA := proc(n) z := padic:-ordp(n, 2); andseq(z=p[2], p in ifactors(n)[2]) end:
%p A367586 select(isA, [seq(1..222)]);  # _Peter Luschny_, Jun 10 2025
%t A367586 Select[Range[100], #==1||EvenQ[#]&&SameQ@@Last/@FactorInteger[#]&]
%Y A367586 Contains all prime powers A000961 and squarefree numbers A005117.
%Y A367586 Partitions of this type (uniform containing 1) are counted by A097986.
%Y A367586 Positions of all one rows {1,1,...} in A367579.
%Y A367586 Positions of powers of 2 in A367580.
%Y A367586 A007947 gives squarefree kernel.
%Y A367586 A027746 lists prime factors, length A001222, indices A112798.
%Y A367586 A027748 lists distinct prime factors, length A001221, indices A304038.
%Y A367586 A071625 counts distinct prime exponents.
%Y A367586 A124010 gives prime signature, sorted A118914.
%Y A367586 A367581 gives multiset multiplicity kernel sum, max A367583, min A055396.
%Y A367586 Cf. A001597, A051904, A072774, A130091, A175781, A367584, A367587, A367685.
%K A367586 nonn
%O A367586 1,2
%A A367586 _Gus Wiseman_, Nov 30 2023