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A367861 Numbers k whose multiset multiplicity cokernel (in which each prime exponent becomes the greatest prime factor with that exponent) is different from that of all positive integers less than k.

%I A367861 #5 Dec 04 2023 06:39:01
%S A367861 1,2,3,5,6,7,10,11,12,13,14,17,19,20,22,23,26,28,29,30,31,34,37,38,41,
%T A367861 42,43,44,45,46,47,52,53,58,59,60,61,62,63,66,67,68,71,73,74,76,78,79,
%U A367861 82,83,84,86,89,90,92,94,97,99,101,102,103,106,107,109,113
%N A367861 Numbers k whose multiset multiplicity cokernel (in which each prime exponent becomes the greatest prime factor with that exponent) is different from that of all positive integers less than k.
%C A367861 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.
%e A367861 The terms together with their prime indices begin:
%e A367861      1: {}         23: {9}        47: {15}
%e A367861      2: {1}        26: {1,6}      52: {1,1,6}
%e A367861      3: {2}        28: {1,1,4}    53: {16}
%e A367861      5: {3}        29: {10}       58: {1,10}
%e A367861      6: {1,2}      30: {1,2,3}    59: {17}
%e A367861      7: {4}        31: {11}       60: {1,1,2,3}
%e A367861     10: {1,3}      34: {1,7}      61: {18}
%e A367861     11: {5}        37: {12}       62: {1,11}
%e A367861     12: {1,1,2}    38: {1,8}      63: {2,2,4}
%e A367861     13: {6}        41: {13}       66: {1,2,5}
%e A367861     14: {1,4}      42: {1,2,4}    67: {19}
%e A367861     17: {7}        43: {14}       68: {1,1,7}
%e A367861     19: {8}        44: {1,1,5}    71: {20}
%e A367861     20: {1,1,3}    45: {2,2,3}    73: {21}
%e A367861     22: {1,5}      46: {1,9}      74: {1,12}
%t A367861 nn=100;
%t A367861 mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367861 qq=Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,nn}];
%t A367861 Select[Range[nn], FreeQ[Take[qq,#-1],qq[[#]]]&]
%Y A367861 Contains all primes A000040 but no other perfect powers A001597.
%Y A367861 All terms are rootless A007916 (have no positive integer roots).
%Y A367861 For kernel instead of cokernel we have A367585, sorted version of A367584.
%Y A367861 The MMC triangle is A367858, sum A367860, min A367857, max A061395.
%Y A367861 Sorted positions of first appearances in A367859.
%Y A367861 A007947 gives squarefree kernel.
%Y A367861 A027746 lists prime factors, length A001222, indices A112798.
%Y A367861 A027748 lists distinct prime factors, length A001221, indices A304038.
%Y A367861 A071625 counts distinct prime exponents.
%Y A367861 A124010 gives prime signature, sorted A118914.
%Y A367861 Cf. A020639, A051904, A072774, A073485, A130091, A181819, A367582, A367768.
%K A367861 nonn
%O A367861 1,2
%A A367861 _Gus Wiseman_, Dec 03 2023