%I #5 Dec 04 2023 06:39:01

%S 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 42,43,44,45,46,47,52,53,58,59,60,61,62,63,66,67,68,71,73,74,76,78,79,

%U 82,83,84,86,89,90,92,94,97,99,101,102,103,106,107,109,113

%N 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 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 The terms together with their prime indices begin:

%e 1: {} 23: {9} 47: {15}

%e 2: {1} 26: {1,6} 52: {1,1,6}

%e 3: {2} 28: {1,1,4} 53: {16}

%e 5: {3} 29: {10} 58: {1,10}

%e 6: {1,2} 30: {1,2,3} 59: {17}

%e 7: {4} 31: {11} 60: {1,1,2,3}

%e 10: {1,3} 34: {1,7} 61: {18}

%e 11: {5} 37: {12} 62: {1,11}

%e 12: {1,1,2} 38: {1,8} 63: {2,2,4}

%e 13: {6} 41: {13} 66: {1,2,5}

%e 14: {1,4} 42: {1,2,4} 67: {19}

%e 17: {7} 43: {14} 68: {1,1,7}

%e 19: {8} 44: {1,1,5} 71: {20}

%e 20: {1,1,3} 45: {2,2,3} 73: {21}

%e 22: {1,5} 46: {1,9} 74: {1,12}

%t nn=100;

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

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

%t Select[Range[nn], FreeQ[Take[qq,#-1],qq[[#]]]&]

%Y Contains all primes A000040 but no other perfect powers A001597.

%Y All terms are rootless A007916 (have no positive integer roots).

%Y For kernel instead of cokernel we have A367585, sorted version of A367584.

%Y The MMC triangle is A367858, sum A367860, min A367857, max A061395.

%Y Sorted positions of first appearances in A367859.

%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. A020639, A051904, A072774, A073485, A130091, A181819, A367582, A367768.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 03 2023