A367584 - OEIS

%I #11 Apr 28 2024 16:21:27

%S 1,2,3,6,5,12,7,30,15,20,11,90,13,28,45,210,17,60,19,150,63,44,23,630,

%T 35,52,105,252,29,360,31,2310,99,68,175,2100,37,76,117,1050,41,504,43,

%U 396,525,92,47,6930,77,140,153,468,53,420,275,1470,171,116,59

%N Least number whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is n. First position of n in A367580.

%C 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 the triangle A367579, and as an operation on their ranks it is represented by A367580.

%F a(p) = p for all primes p.

%e The least number with multiset multiplicity kernel 9 is 15, so a(9) = 15.

%e The terms together with their prime indices begin:

%e    1 ->  1: {}

%e    2 ->  2: {1}

%e    3 ->  3: {2}

%e    4 ->  6: {1,2}

%e    5 ->  5: {3}

%e    6 -> 12: {1,1,2}

%e    7 ->  7: {4}

%e    8 -> 30: {1,2,3}

%e    9 -> 15: {2,3}

%e   10 -> 20: {1,1,3}

%e   11 -> 11: {5}

%e   12 -> 90: {1,2,2,3}

%e   13 -> 13: {6}

%e   14 -> 28: {1,1,4}

%e   15 -> 45: {2,2,3}

%e   16 ->210: {1,2,3,4}

%t nn=1000;

%t mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];

%t spnm[y_]:=Max@@NestWhile[Most, Sort[y], Union[#]!=Range[Max@@#]&];

%t qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,nn}];

%t Table[Position[qq,i][[1,1]], {i,spnm[qq]}]

%Y Positions of primes are A000040.

%Y Positions of squarefree numbers are A000961.

%Y All terms are rootless A007916.

%Y Contains no nonprime prime powers A246547.

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

%Y Positions of first appearances in A367580.

%Y The sorted version is A367585.

%Y The complement is A367768.

%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 Cf. A005117, A020639, A051904, A072774, A130091, A367586.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 29 2023