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%I #5 Dec 04 2023 06:39:21
%S 1,2,3,2,5,9,7,2,3,25,11,6,13,49,25,2,17,6,19,10,49,121,23,6,5,169,3,
%T 14,29,125,31,2,121,289,49,9,37,361,169,10,41,343,43,22,15,529,47,6,7,
%U 10,289,26,53,6,121,14,361,841,59,50,61,961,21,2,169,1331
%N Multiset multiplicity cokernel (MMC) of n. Product of (greatest 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 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 cokernel MMC(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 max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.
%F a(n^k) = a(n) for all positive integers n and k.
%F If n is squarefree, a(n) = A006530(n)^A001222(n).
%F A055396(a(n)) = A367587(n).
%F A056239(a(n)) = A367860(n).
%F A061395(a(n)) = A061395(n).
%F A001222(a(n)) = A001221(n).
%F A001221(a(n)) = A071625(n).
%F A071625(a(n)) = A323022(n).
%e 90 has prime factorization 2^1*3^2*5^1, so for k = 1 we have 5^2, and for k = 2 we have 3^1, so a(90) = 75.
%t mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,30}]
%Y Positions of 2's are A000079 without 1.
%Y Positions of 3's are A000244 without 1.
%Y Positions of primes (including 1) are A000961.
%Y Depends only on rootless base A052410, see A007916.
%Y Positions of prime powers are A072774.
%Y Positions of squarefree numbers are A130091.
%Y For kernel instead of cokernel we have A367580, ranks of A367579.
%Y Rows of A367858 have this rank, sum A367860, max A061395, min A367587.
%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 multiset of multiplicities (prime signature), sorted A118914.
%Y Cf. A005117, A020639, A052409, A175781, A181819, A353742, A367584, A367585.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 03 2023
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