A367585 Numbers k whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is different from that of all positive integers less than k.

1, 2, 3, 5, 6, 7, 11, 12, 13, 15, 17, 19, 20, 23, 28, 29, 30, 31, 35, 37, 41, 43, 44, 45, 47, 52, 53, 59, 60, 61, 63, 67, 68, 71, 73, 76, 77, 79, 83, 89, 90, 92, 97, 99, 101, 103, 105, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 140, 143, 148, 149, 150

Offset

1,2

Comments

We define the multiset multiplicity kernel (MMK) of a positive integer n to be the 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. For example, 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 MMK(90) = 12. As an operation on multisets, MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

Links

Table of n, a(n) for n=1..62.

Example

The terms together with their prime indices begin:
     1: {}         28: {1,1,4}    60: {1,1,2,3}
     2: {1}        29: {10}       61: {18}
     3: {2}        30: {1,2,3}    63: {2,2,4}
     5: {3}        31: {11}       67: {19}
     6: {1,2}      35: {3,4}      68: {1,1,7}
     7: {4}        37: {12}       71: {20}
    11: {5}        41: {13}       73: {21}
    12: {1,1,2}    43: {14}       76: {1,1,8}
    13: {6}        44: {1,1,5}    77: {4,5}
    15: {2,3}      45: {2,2,3}    79: {22}
    17: {7}        47: {15}       83: {23}
    19: {8}        52: {1,1,6}    89: {24}
    20: {1,1,3}    53: {16}       90: {1,2,2,3}
    23: {9}        59: {17}       92: {1,1,9}

Mathematica

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

Crossrefs

Contains all primes A000040 but no other perfect powers A001597.

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

Positions of squarefree terms appear to be A073485.

Contains no non-prime prime powers A246547.

The MMK triangle is A367579, sum A367581, min A055396, max A367583.

Sorted positions of first appearances in A367580.

Sorted version of A367584.

Complement of A367768.

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, A130091, A181819, A238747, A367582, A367685.

Sequence in context: A059041 A129128 A355647 * A164922 A205523 A343027

Adjacent sequences: A367582 A367583 A367584 * A367586 A367587 A367588

Keyword

nonn

Author

Gus Wiseman, Nov 29 2023

Status

approved