A367586 - OEIS

%I #9 Dec 04 2023 08:32:51

%S 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 82,86,94,100,102,106,110,114,118,122,128,130,134,138,142,146,154,158,

%U 166,170,174,178,182,186,190,194,196,202,206,210,214,216,218,222

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

%e We have MMK({1,1,2,2}) = {1,1} so 36 is in the sequence.

%e The terms together with their prime indices begin:

%e     1: {}

%e     2: {1}

%e     4: {1,1}

%e     6: {1,2}

%e     8: {1,1,1}

%e    10: {1,3}

%e    14: {1,4}

%e    16: {1,1,1,1}

%e    22: {1,5}

%e    26: {1,6}

%e    30: {1,2,3}

%e    32: {1,1,1,1,1}

%e    34: {1,7}

%e    36: {1,1,2,2}

%e    38: {1,8}

%e    42: {1,2,4}

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

%Y Contains all prime powers A000961 and squarefree numbers A005117.

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

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

%Y Positions of powers of 2 in A367580.

%Y A007947 gives squarefree kernel.

%Y A027746 lists prime factors, length A001222, indices A112798.

%Y A027748 lists distinct prime factors, length A001221, indices A304038.

%Y A071625 counts distinct prime exponents.

%Y A124010 gives prime signature, sorted A118914.

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 30 2023