A367768 -id:A367768 - cp's OEIS frontend

A367584 Least number whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is n. First position of n in A367580.

1, 2, 3, 6, 5, 12, 7, 30, 15, 20, 11, 90, 13, 28, 45, 210, 17, 60, 19, 150, 63, 44, 23, 630, 35, 52, 105, 252, 29, 360, 31, 2310, 99, 68, 175, 2100, 37, 76, 117, 1050, 41, 504, 43, 396, 525, 92, 47, 6930, 77, 140, 153, 468, 53, 420, 275, 1470, 171, 116, 59

Offset: 1

Gus Wiseman, Nov 29 2023

nonn

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 the triangle A367579, and as an operation on their ranks it is represented by A367580.

			The least number with multiset multiplicity kernel 9 is 15, so a(9) = 15.
The terms together with their prime indices begin:
   1 ->  1: {}
   2 ->  2: {1}
   3 ->  3: {2}
   4 ->  6: {1,2}
   5 ->  5: {3}
   6 -> 12: {1,1,2}
   7 ->  7: {4}
   8 -> 30: {1,2,3}
   9 -> 15: {2,3}
  10 -> 20: {1,1,3}
  11 -> 11: {5}
  12 -> 90: {1,2,2,3}
  13 -> 13: {6}
  14 -> 28: {1,1,4}
  15 -> 45: {2,2,3}
  16 ->210: {1,2,3,4}

Positions of primes are A000040.
Positions of squarefree numbers are A000961.
All terms are rootless A007916.
Contains no nonprime prime powers A246547.
The MMK triangle is A367579, sum A367581, min A055396, max A367583.
Positions of first appearances in A367580.
The sorted version is A367585.
The complement is 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. A005117, A020639, A051904, A072774, A130091, A367586.

nn=1000;
mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
spnm[y_]:=Max@@NestWhile[Most, Sort[y], Union[#]!=Range[Max@@#]&];
qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,nn}];
Table[Position[qq,i][[1,1]], {i,spnm[qq]}]

a(p) = p for all primes p.

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.

Original entry on oeis.org

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

Gus Wiseman, Nov 29 2023

nonn

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.

			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}

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 nonprime 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.

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[[#]]]&]

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.

Original entry on oeis.org

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

Gus Wiseman, Dec 03 2023

nonn

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.

			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}

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.

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[[#]]]&]

cp's OEIS Frontend

A367584 Least number whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is n. First position of n in A367580.

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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.

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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.

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Mathematica