cp's OEIS Frontend

A367580 Multiset multiplicity kernel (MMK) of n. 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.

%I A367580 #15 Dec 04 2023 08:32:56
%S A367580 1,2,3,2,5,4,7,2,3,4,11,6,13,4,9,2,17,6,19,10,9,4,23,6,5,4,3,14,29,8,
%T A367580 31,2,9,4,25,4,37,4,9,10,41,8,43,22,15,4,47,6,7,10,9,26,53,6,25,14,9,
%U A367580 4,59,18,61,4,21,2,25,8,67,34,9,8,71,6,73,4,15,38
%N A367580 Multiset multiplicity kernel (MMK) of n. 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.
%C A367580 As an operation on multisets, this is represented by A367579.
%F A367580 a(n^k) = a(n) for all positive integers n and k.
%F A367580 A001221(a(n)) = A071625(n).
%F A367580 A001222(a(n)) = A001221(n).
%F A367580 If n is squarefree, a(n) = A020639(n)^A001222(n).
%F A367580 A056239(a(n)) = A367581(n).
%e A367580 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 a(90) = 12.
%t A367580 mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367580 Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,100}]
%Y A367580 Positions of 2's are A000079 without 1.
%Y A367580 Positions of 3's are A000244 without 1.
%Y A367580 Positions of primes (including 1) are A000961.
%Y A367580 Positions of prime(k) are prime powers prime(k)^i, rows of A051128.
%Y A367580 Depends only on rootless base A052410, see A007916.
%Y A367580 Positions of prime powers are A072774.
%Y A367580 Positions of squarefree numbers are A130091.
%Y A367580 Agrees with A181819 at positions A367683, counted by A367682.
%Y A367580 Rows of A367579 have this rank, sum A367581, max A367583, min A055396.
%Y A367580 Positions of first appearances are A367584, sorted A367585.
%Y A367580 Positions of powers of 2 are A367586.
%Y A367580 Divides n at positions A367685, counted by A367684.
%Y A367580 The opposite version (cokernel) is A367859.
%Y A367580 A007947 gives squarefree kernel.
%Y A367580 A027746 lists prime factors, length A001222, indices A112798.
%Y A367580 A027748 lists distinct prime factors, length A001221, indices A304038.
%Y A367580 A071625 counts distinct prime exponents.
%Y A367580 A124010 gives multiset of multiplicities (prime signature), sorted A118914.
%Y A367580 Cf. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.
%K A367580 nonn
%O A367580 1,2
%A A367580 _Gus Wiseman_, Nov 26 2023