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.

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, 42, 43, 44, 45, 46, 47, 52, 53, 58, 59, 60, 61, 62, 63, 66, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 84, 86, 89, 90, 92, 94, 97, 99, 101, 102, 103, 106, 107, 109, 113

Offset: 1

Author

Gus Wiseman, Dec 03 2023

Keywords

• nonn

Comments

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.

Examples

			The terms together with their prime indices begin:
     1: {}         23: {9}        47: {15}
     2: {1}        26: {1,6}      52: {1,1,6}
     3: {2}        28: {1,1,4}    53: {16}
     5: {3}        29: {10}       58: {1,10}
     6: {1,2}      30: {1,2,3}    59: {17}
     7: {4}        31: {11}       60: {1,1,2,3}
    10: {1,3}      34: {1,7}      61: {18}
    11: {5}        37: {12}       62: {1,11}
    12: {1,1,2}    38: {1,8}      63: {2,2,4}
    13: {6}        41: {13}       66: {1,2,5}
    14: {1,4}      42: {1,2,4}    67: {19}
    17: {7}        43: {14}       68: {1,1,7}
    19: {8}        44: {1,1,5}    71: {20}
    20: {1,1,3}    45: {2,2,3}    73: {21}
    22: {1,5}      46: {1,9}      74: {1,12}
		

Crossrefs

Contains all primes A000040 but no other perfect powers A001597.

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

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

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

Sorted positions of first appearances in A367859.

A007947 gives squarefree kernel.

A027746 lists prime factors, length A001222, indices A112798.

A027748 lists distinct prime factors, length A001221, indices A304038.

A071625 counts distinct prime exponents.

A124010 gives prime signature, sorted A118914.

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

Programs

Mathematica

nn=100;
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
qq=Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,nn}];
Select[Range[nn], FreeQ[Take[qq,#-1],qq[[#]]]&]