A367585 -id:A367585 - cp's OEIS frontend

A367580 Multiset multiplicity kernel (MMK) of n. 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.

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 6, 13, 4, 9, 2, 17, 6, 19, 10, 9, 4, 23, 6, 5, 4, 3, 14, 29, 8, 31, 2, 9, 4, 25, 4, 37, 4, 9, 10, 41, 8, 43, 22, 15, 4, 47, 6, 7, 10, 9, 26, 53, 6, 25, 14, 9, 4, 59, 18, 61, 4, 21, 2, 25, 8, 67, 34, 9, 8, 71, 6, 73, 4, 15, 38

Offset: 1

Gus Wiseman, Nov 26 2023

nonn

As an operation on multisets, this is represented by A367579.

			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 a(90) = 12.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Positions of prime(k) are prime powers prime(k)^i, rows of A051128.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
Agrees with A181819 at positions A367683, counted by A367682.
Rows of A367579 have this rank, sum A367581, max A367583, min A055396.
Positions of first appearances are A367584, sorted A367585.
Positions of powers of 2 are A367586.
Divides n at positions A367685, counted by A367684.
The opposite version (cokernel) is 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 multiset of multiplicities (prime signature), sorted A118914.
Cf. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.

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

a(n^k) = a(n) for all positive integers n and k.
A001221(a(n)) = A071625(n).
A001222(a(n)) = A001221(n).
If n is squarefree, a(n) = A020639(n)^A001222(n).
A056239(a(n)) = A367581(n).

A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 6, 1, 1, 2, 2, 1, 7, 1, 2, 8, 1, 3, 2, 2, 1, 1, 9, 1, 2, 3, 1, 1, 2, 1, 4, 10, 1, 1, 1, 11, 1, 2, 2, 1, 1, 3, 3, 1, 1, 12, 1, 1, 2, 2, 1, 3, 13, 1, 1, 1, 14, 1, 5, 2, 3, 1, 1, 15, 1, 2, 4, 1, 3, 2, 2, 1, 6, 16, 1, 2

Offset: 1

Gus Wiseman, Nov 25 2023

Row n = 1 is empty.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the multiset multiplicity kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.
Note: I chose the word 'kernel' because, as with A007947 and A304038, MMK(m) is constructed using the same underlying elements as m and has length equal to the number of distinct elements of m. However, it is not necessarily a submultiset of m.

			The first 45 rows:
     1: {}      16: {1}       31: {11}
     2: {1}     17: {7}       32: {1}
     3: {2}     18: {1,2}     33: {2,2}
     4: {1}     19: {8}       34: {1,1}
     5: {3}     20: {1,3}     35: {3,3}
     6: {1,1}   21: {2,2}     36: {1,1}
     7: {4}     22: {1,1}     37: {12}
     8: {1}     23: {9}       38: {1,1}
     9: {2}     24: {1,2}     39: {2,2}
    10: {1,1}   25: {3}       40: {1,3}
    11: {5}     26: {1,1}     41: {13}
    12: {1,2}   27: {2}       42: {1,1,1}
    13: {6}     28: {1,4}     43: {14}
    14: {1,1}   29: {10}      44: {1,5}
    15: {2,2}   30: {1,1,1}   45: {2,3}

Indices of empty and singleton rows are A000961.
Row lengths are A001221.
Depends only on rootless base A052410, see A007916.
Row minima are A055396.
Rows have A071625 distinct elements.
Indices of constant rows are A072774.
Indices of strict rows are A130091.
Rows have Heinz numbers A367580.
Row sums are A367581.
Row maxima are A367583, opposite A367587.
Index of first row with Heinz number n is A367584.
Sorted row indices of first appearances are A367585.
Indices of rows of the form {1,1,...} are A367586.
Agrees with sorted prime signature at A367683, counted by A367682.
A submultiset of prime indices at A367685, counted by A367684.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]

For all positive integers n and k, row n^k is the same as row n.

A367581 Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 3, 6, 2, 4, 1, 7, 3, 8, 4, 4, 2, 9, 3, 3, 2, 2, 5, 10, 3, 11, 1, 4, 2, 6, 2, 12, 2, 4, 4, 13, 3, 14, 6, 5, 2, 15, 3, 4, 4, 4, 7, 16, 3, 6, 5, 4, 2, 17, 5, 18, 2, 6, 1, 6, 3, 19, 8, 4, 3, 20, 3, 21, 2, 5, 9, 8, 3, 22, 4, 2, 2

Offset: 1

Gus Wiseman, Nov 28 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define the multiset multiplicity kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.

			The multiset multiplicity kernel of {1,2,2,3} is {1,1,2}, so a(90) = 4.

Positions of 1's are A000079 without 1.
Positions of first appearances are A008578.
Depends only on rootless base A052410, see A007916, A052409.
The triangle A367579 has these as row sums, ranks A367580.
The triangle for this rank statistic is A367582.
For maximum instead of sum we have A367583, opposite A367587.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
Cf. A000720, A005117, A051904, A055396, A061395, A071625, A072774, A130091, A175781, A367584, A367585.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Total[mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[n]]], {n,100}]

a(n^k) = a(n) for all positive integers n and k.
a(n) = A056239(A367580(n)).
If n is squarefree, a(n) = A055396(n)*A001222(n).

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.

Original entry on oeis.org

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.

A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 2, 8, 3, 2, 1, 9, 2, 3, 1, 2, 4, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 4, 3, 2, 6, 16, 2, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 3, 8, 4, 1, 22, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 28 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define the multiset multiplicity kernel MMK(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 min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.

			For 450 = 2^1 * 3^2 * 5^2, we have MMK({1,2,2,3,3}) = {1,2,2} so a(450) = 2.

Positions of first appearances are A008578.
Depends only on rootless base A052410, see A007916, A052409.
For minimum instead of maximum element we have A055396.
Row maxima of A367579.
Greatest prime index of A367580.
Positions of 1's are A367586 (powers of even squarefree numbers).
The opposite version is A367587.
A007947 gives squarefree kernel.
A072774 lists powers of squarefree numbers.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A363486 gives least prime index of greatest exponent.
A363487 gives greatest prime index of greatest exponent.
A364191 gives least prime index of least exponent.
A364192 gives greatest prime index of least exponent.
Cf. A000720, A000961, A051904, A061395, A071625, A130091, A289023, A367581, A367584, A367585, A367683.

mmk[q_]:=With[{mts=Length/@Split[q]},Sort[Table[Min@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Max@@mmk[PrimePi/@Join@@ConstantArray@@@If[n==1,{},FactorInteger[n]]]],{n,1,100}]

a(n) = A061395(A367580(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A055396(n).

A367587 Least element in row n of A367858 (multiset multiplicity cokernel).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 1, 6, 4, 3, 1, 7, 1, 8, 1, 4, 5, 9, 1, 3, 6, 2, 1, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 1, 13, 4, 14, 1, 2, 9, 15, 1, 4, 1, 7, 1, 16, 1, 5, 1, 8, 10, 17, 1, 18, 11, 2, 1, 6, 5, 19, 1, 9, 4, 20, 1, 21, 12, 2, 1, 5, 6, 22, 1, 2

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.

Indices of first appearances are A008578.
Depends only on rootless base A052410, see A007916.
For kernel instead of cokernel we have A055396.
For maximum instead of minimum element we have A061395.
The opposite version is A367583.
Row-minima of A367858.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, sorted A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367579 lists MMK, rank A367580, sum A367581, max A367583, min A055396.
Cf. A000720, A027746, A051904, A071625, A072774, A130091, A367582, A367584, A367585, A367586.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Min@@mmc[prix[n]]],{n,100}]

a(n) = A055396(A367859(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A061395(n).

A367859 Multiset multiplicity cokernel (MMC) of n. Product of (greatest prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 9, 7, 2, 3, 25, 11, 6, 13, 49, 25, 2, 17, 6, 19, 10, 49, 121, 23, 6, 5, 169, 3, 14, 29, 125, 31, 2, 121, 289, 49, 9, 37, 361, 169, 10, 41, 343, 43, 22, 15, 529, 47, 6, 7, 10, 289, 26, 53, 6, 121, 14, 361, 841, 59, 50, 61, 961, 21, 2, 169, 1331

Offset: 1

Gus Wiseman, Dec 03 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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.

			90 has prime factorization 2^1*3^2*5^1, so for k = 1 we have 5^2, and for k = 2 we have 3^1, so a(90) = 75.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
For kernel instead of cokernel we have A367580, ranks of A367579.
Rows of A367858 have this rank, sum A367860, max A061395, min A367587.
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 multiset of multiplicities (prime signature), sorted A118914.
Cf. A005117, A020639, A052409, A175781, A181819, A353742, A367584, A367585.

mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,30}]

a(n^k) = a(n) for all positive integers n and k.
If n is squarefree, a(n) = A006530(n)^A001222(n).
A055396(a(n)) = A367587(n).
A056239(a(n)) = A367860(n).
A061395(a(n)) = A061395(n).
A001222(a(n)) = A001221(n).
A001221(a(n)) = A071625(n).
A071625(a(n)) = A323022(n).

A367858 Irregular triangle read by rows where row n is the multiset multiplicity cokernel (MMC) of the multiset of prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 1, 2, 3, 3, 5, 1, 2, 6, 4, 4, 3, 3, 1, 7, 1, 2, 8, 1, 3, 4, 4, 5, 5, 9, 1, 2, 3, 6, 6, 2, 1, 4, 10, 3, 3, 3, 11, 1, 5, 5, 7, 7, 4, 4, 2, 2, 12, 8, 8, 6, 6, 1, 3, 13, 4, 4, 4, 14, 1, 5, 2, 3, 9, 9, 15, 1, 2, 4, 1, 3, 7, 7, 1, 6, 16, 1, 2

Offset: 1

Gus Wiseman, Dec 03 2023

Row n = 1 is empty.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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}.

			The first 45 rows:
     1: {}       16: {1}        31: {11}
     2: {1}      17: {7}        32: {1}
     3: {2}      18: {1,2}      33: {5,5}
     4: {1}      19: {8}        34: {7,7}
     5: {3}      20: {1,3}      35: {4,4}
     6: {2,2}    21: {4,4}      36: {2,2}
     7: {4}      22: {5,5}      37: {12}
     8: {1}      23: {9}        38: {8,8}
     9: {2}      24: {1,2}      39: {6,6}
    10: {3,3}    25: {3}        40: {1,3}
    11: {5}      26: {6,6}      41: {13}
    12: {1,2}    27: {2}        42: {4,4,4}
    13: {6}      28: {1,4}      43: {14}
    14: {4,4}    29: {10}       44: {1,5}
    15: {3,3}    30: {3,3,3}    45: {2,3}

Indices of empty and singleton rows are A000961.
Row lengths are A001221.
Depends only on rootless base A052410, see A007916.
Row maxima are A061395.
Rows have A071625 distinct elements.
Indices of constant rows are A072774.
Indices of strict rows are A130091.
Row minima are A367587.
Rows have Heinz numbers A367859.
Row sums are A367860.
Sorted row indices of first appearances are A367861, for kernel A367585.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
Cf. A000720, A027746, A051904, A052409, A061395, A175781, A367582, A367583.

mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[mmc[PrimePi /@ Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]

For all positive integers n and k, row n^k is the same as row n.

A367768 Numbers k such that MMK(k) = MMK(i) for some i < k, where MMK is multiset multiplicity kernel A367580.

Original entry on oeis.org

4, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 27, 32, 33, 34, 36, 38, 39, 40, 42, 46, 48, 49, 50, 51, 54, 55, 56, 57, 58, 62, 64, 65, 66, 69, 70, 72, 74, 75, 78, 80, 81, 82, 84, 85, 86, 87, 88, 91, 93, 94, 95, 96, 98, 100, 102, 104, 106, 108, 110, 111, 112, 114

Offset: 1

Gus Wiseman, Dec 01 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:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   32: {1,1,1,1,1}
   33: {2,5}
   34: {1,7}
   36: {1,1,2,2}

The squarefree case is A073486, complement A073485.
The MMK triangle is A367579, sum A367581, min A055396, max A367583.
Sorted positions of non-first appearances in A367580.
The complement is A367585, sorted version of A367584.
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. A051904, A072774, A130091, A181819, A246547, 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], MemberQ[Take[qq,#-1], qq[[#]]]&]

A367580(a(k)) = A367580(i) for some i < a(k).

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

A367580 Multiset multiplicity kernel (MMK) of n. 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.

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A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

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A367581 Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.

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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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A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

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A367587 Least element in row n of A367858 (multiset multiplicity cokernel).

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A367859 Multiset multiplicity cokernel (MMC) of n. Product of (greatest prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

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A367858 Irregular triangle read by rows where row n is the multiset multiplicity cokernel (MMC) of the multiset of prime indices of n.

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