A304038 -id:A304038 - cp's OEIS frontend

A372430 Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.

1, 3, 5, 15, 27, 39, 55, 63, 85, 121, 125, 135, 169, 171, 175, 209, 243, 247, 255, 299, 375, 399, 437, 459, 507, 539, 605, 637, 725, 735, 783, 841, 867, 891, 1085, 1215, 1323, 1331, 1375, 1519, 1767, 1815, 1863, 2079, 2125, 2187, 2223, 2295, 2299, 2331, 2405

Offset: 1

Gus Wiseman, May 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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.
Conjecture: The only number whose binary indices are a subset of its prime indices is 4100, with binary indices {3,13} and prime indices {1,1,3,3,13}. Verified up to 10,000,000.

			The prime indices of 135 are {2,2,2,3}, and the binary indices are {1,2,3,8}. Since {2,3} is a subset of {1,2,3,8}, 135 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     3: {2}
     5: {3}
    15: {2,3}
    27: {2,2,2}
    39: {2,6}
    55: {3,5}
    63: {2,2,4}
    85: {3,7}
   121: {5,5}
   125: {3,3,3}
The terms together with their binary expansions and binary indices begin:
     1:              1 ~ {1}
     3:             11 ~ {1,2}
     5:            101 ~ {1,3}
    15:           1111 ~ {1,2,3,4}
    27:          11011 ~ {1,2,4,5}
    39:         100111 ~ {1,2,3,6}
    55:         110111 ~ {1,2,3,5,6}
    63:         111111 ~ {1,2,3,4,5,6}
    85:        1010101 ~ {1,3,5,7}
   121:        1111001 ~ {1,4,5,6,7}
   125:        1111101 ~ {1,3,4,5,6,7}

The version for equal lengths is A071814, zeros of A372441.
The version for equal sums is A372427, zeros of A372428.
For disjoint instead of subset we have A372431, complement A372432.
The version for equal maxima is A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A001221, A230877, A243055, A304818, A355536, A358136, A372429.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SubsetQ[bix[#],prix[#]]&]

Row k of A304038 is a subset of row k of A048793.

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.

A381076 Sorted positions of first appearances in A066503 (n minus squarefree kernel of n).

Original entry on oeis.org

1, 4, 8, 16, 18, 20, 24, 25, 27, 32, 44, 48, 50, 52, 54, 64, 68, 72, 75, 76, 80, 81, 92, 96, 98, 108, 112, 116, 121, 125, 128, 144, 148, 152, 160, 162, 164, 172, 175, 176, 188, 189, 192, 196, 198, 200, 212, 216, 232, 236, 242, 243, 244, 256, 260, 264, 268, 272

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

In A066503, each value appears for the first time at one of these positions.

For quotient instead of difference we have A001694, sorted firsts of A003557.
Sorted positions of first appearances in A066503.
For indices and sum we have A380957 (unsorted A380956), firsts of A380955.
For indices and quotient we have A380988 (unsorted A380987), firsts of A290106.
For sum instead of product we have A381075, sorted firsts of A280292, see A280286.
For indices instead of factors we have A381077, sorted firsts of A380986.
A000040 lists the primes, differences A001223.
A001414 adds up prime factors (indices A056239), row sums of A027746 (indices A112798).
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
Cf. A001221, A001222, A038838, A046660, A075255, A081770, A116861, A136565, A178503, A175508, A304038, A380989.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Times@@prifacs[n]-Times@@Union[prifacs[n]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A365003 Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.

Original entry on oeis.org

1, 4, 9, 25, 36, 48, 49, 100, 121, 160, 169, 196, 225, 289, 361, 441, 448, 484, 529, 567, 676, 750, 810, 841, 900, 961, 1080, 1089, 1156, 1200, 1225, 1369, 1408, 1440, 1444, 1521, 1681, 1764, 1849, 1920, 2116, 2209, 2268, 2352, 2601, 2809, 3024, 3025, 3159

Offset: 1

Gus Wiseman, Aug 23 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 750 are {1,2,3,3,3}, with sum 12, while the distinct prime indices {1,2,3} have sum 6, so 750 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    25: {3,3}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    49: {4,4}
   100: {1,1,3,3}
   121: {5,5}
   160: {1,1,1,1,1,3}
   169: {6,6}
   196: {1,1,4,4}
   225: {2,2,3,3}
   289: {7,7}
   361: {8,8}
   441: {2,2,4,4}
   448: {1,1,1,1,1,1,4}

The LHS is A056239 (sum of prime indices).
The RHS is twice A066328.
Partitions of this type are counted by A364910.
A000041 counts integer partitions, strict A000009.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, distinct A304038.
A116861 counts partitions by sum and sum of distinct parts.
A323092 counts double-free partitions, ranks A320340.
Cf. A364350, A364839, A364906, A364907, A364911, A364916.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[prix[#]]==2*Total[Union[prix[#]]]&]

A056239(a(n)) = 2*A066328(a(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).

A367860 Sum of the multiset multiplicity cokernel (in which each multiplicity becomes the greatest element of that multiplicity) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 4, 1, 2, 6, 5, 3, 6, 8, 6, 1, 7, 3, 8, 4, 8, 10, 9, 3, 3, 12, 2, 5, 10, 9, 11, 1, 10, 14, 8, 4, 12, 16, 12, 4, 13, 12, 14, 6, 5, 18, 15, 3, 4, 4, 14, 7, 16, 3, 10, 5, 16, 20, 17, 7, 18, 22, 6, 1, 12, 15, 19, 8, 18, 12, 20, 3, 21, 24, 5, 9, 10

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.

			The multiset multiplicity cokernel of {1,2,2,3} is {2,3,3}, so a(90) = 8.

Positions of 1's are A000079 without 1.
Depends only on rootless base A052410, see A007916, A052409.
For kernel instead of cokernel we have A367581, row-sums of A367579.
For minimum instead of sum we have A367587, opposite A367583.
The triangle A367858 has these as row sums, ranks A367859.
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, A051904, A055396, A061395, A071625, A072774, A130091, A367580, A175781, A367582, A367584.

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

A329025 If n = Product (p_j^k_j) then a(n) = concatenation (pi(p_j)), where pi = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 12, 4, 1, 2, 13, 5, 12, 6, 14, 23, 1, 7, 12, 8, 13, 24, 15, 9, 12, 3, 16, 2, 14, 10, 123, 11, 1, 25, 17, 34, 12, 12, 18, 26, 13, 13, 124, 14, 15, 23, 19, 15, 12, 4, 13, 27, 16, 16, 12, 35, 14, 28, 110, 17, 123, 18, 111, 24, 1, 36, 125, 19, 17, 29, 134

Offset: 1

Ilya Gutkovskiy, Nov 02 2019

Concatenate of indices of distinct prime factors of n, in increasing order.

			a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 123.

Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A037916, A080695, A084317, A127668, A304038, A329026.

a[n_] := FromDigits[Flatten@IntegerDigits@(PrimePi[#[[1]]] & /@ FactorInteger[n])]; Table[a[n], {n, 1, 70}]

a(prime(n)^k) = n for k > 0.

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 4, 1, 4, 2, 4, 3, 4, 1, 2, 4, 1, 3, 4, 2, 3, 4, 1, 2, 3, 4, 5, 1, 5, 2, 5, 3, 5, 4, 5, 1, 2, 5, 1, 3, 5, 1, 4, 5, 2, 3, 5, 2, 4, 5, 3, 4, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 49, 63, 81, 99, 121, 125, 135, 169, 171, 245, 279, 289, 343, 361, 363, 369, 375, 387, 477, 529, 531, 575, 603, 625, 675, 711, 729, 747, 833, 841, 847, 873, 875, 891, 909, 961, 981, 1029, 1083, 1125, 1127, 1179, 1225, 1251, 1377, 1413, 1445, 1467

Offset: 1

Gus Wiseman, Feb 20 2025

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. A position of first appearance in a sequence q is an index k such that q(k) is different from q(j) for all j < k.

All terms are odd.

			The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    63: {2,2,4}
    81: {2,2,2,2}
    99: {2,2,5}
   121: {5,5}
   125: {3,3,3}
   135: {2,2,2,3}
   169: {6,6}
   171: {2,2,8}
   245: {3,4,4}
   279: {2,2,11}

For length instead of product we have A151821, firsts of A046660.
For factors instead of indices we have A381076, sorted firsts of A066503.
For sum of factors instead of product of indices we have A381075 (unsorted A280286), A280292.
For quotient instead of difference we have A380988 (unsorted A380987), firsts of A290106.
For quotient and factors we have A001694 (unsorted A064549), firsts of A003557.
For sum instead of product we have A380957 (unsorted A380956), firsts of A380955.
Sorted firsts of A380986, which has nonzero terms at positions A038838.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000079, A000720, A001222, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]-Times@@Union[prix[n]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

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

A372430 Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.

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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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A381076 Sorted positions of first appearances in A066503 (n minus squarefree kernel of n).

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A365003 Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.

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

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A367860 Sum of the multiset multiplicity cokernel (in which each multiplicity becomes the greatest element of that multiplicity) of the prime indices of n.

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A329025 If n = Product (p_j^k_j) then a(n) = concatenation (pi(p_j)), where pi = A000720.

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A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

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A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

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