A355733 -id:A355733 - cp's OEIS frontend

A355537 Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 8, 16, 32, 32, 32, 64, 64, 128, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 12288, 12288, 12288, 24576, 49152, 98304, 196608, 196608, 393216, 786432, 1572864, 1572864, 4718592, 4718592, 9437184, 18874368, 37748736

Offset: 1

Gus Wiseman, Jul 20 2022

nonn

Also partial products of A001221 without the first term 0, sum A013939.

For initial terms up to n = 29 we have a(n) = 2^A355538(n). The first non-power of 2 is a(30) = 12288.

			The a(n) choices for n = 2, 6, 10, 12, with prime(k) replaced by k:
  (1)  (12131)  (121314121)  (12131412151)
       (12132)  (121314123)  (12131412152)
                (121324121)  (12131412351)
                (121324123)  (12131412352)
                             (12132412151)
                             (12132412152)
                             (12132412351)
                             (12132412352)

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
The version for divisors instead of prime factors is A066843.
The integers themselves are the rows of A131818.
The version with multiplicity is A327486.
Using prime indices instead of 2..n gives A355741, for multisets A355744.
Counting sequences instead of multisets gives A355746.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
Cf. A000005, A000040, A000720, A002110, A013939, A076610, A355538, A355731, A355733, A355745, A355747.

Table[Times@@PrimeNu/@Range[2,m],{m,2,30}]

A355748 Number of ways to choose a sequence of divisors, one of each part of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 2, 2, 2, 1, 2, 3, 4, 2, 4, 4, 4, 2, 3, 2, 4, 2, 2, 2, 2, 1, 4, 2, 6, 3, 4, 4, 4, 2, 6, 4, 8, 4, 4, 4, 4, 2, 2, 3, 4, 2, 4, 4, 4, 2, 3, 2, 4, 2, 2, 2, 2, 1, 2, 4, 4, 2, 6, 6, 6, 3, 6, 4, 8, 4, 4, 4, 4, 2, 4, 6, 8, 4, 8, 8, 8

Offset: 0

Gus Wiseman, Jul 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition number 152 in standard order is (3,1,4), and the a(152) = 6 choices are: (1,1,1), (1,1,2), (1,1,4), (3,1,1), (3,1,2), (3,1,4).

Positions of 1's are A000079 (after the first).
The anti-run case is A354578, zeros A354904, firsts A354905.
An unordered version (using prime indices) is A355731:
- firsts A355732,
- resorted A355733,
- weakly increasing A355735,
- relatively prime A355737,
- strict A355739.
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A005811 counts runs in binary expansion.
A029837 adds up standard compositions, lengths A000120.
A066099 lists the compositions in standard order.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
Cf. A124767, A175413, A238279, A274174, A326841, A333381, A333755, A353847, A353849, A355747.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Times@@Length/@Divisors/@stc[n],{n,0,100}]

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 37, 37, 38, 39, 39, 40, 42, 42, 43, 44, 46, 46

Offset: 1

Gus Wiseman, Jul 23 2022

nonn

For initial terms up to 30 we have a(n) = Log_2 A355537(n).

The sum of the same range is A000096.
The product of the same range is A000142, Heinz number A070826.
For divisors (not just prime factors) we get A002541, also A006218, A077597.
A shifted variation is A013939.
The unshifted version is A022559, product A327486, w/o multiplicity A355537.
The ranges themselves are the rows of A131818 (shifted).
Partial sums of A297155 (shifted).
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A066843 gives partial sums of A000005.
Cf. A000720, A002110, A076610, A305054, A355538, A355731, A355733, A355741, A355744, A355745, A355746, A355747.

Table[Total[(PrimeNu[#]-1)&/@Range[2,n]],{n,1,100}]

a(n) = A013939(n) - n + 1.

A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

1, 2, 6, 9, 15, 49, 35, 27, 45, 98, 63, 105, 171, 117, 81, 135, 245, 343, 273, 549, 189, 1083, 315, 5618, 741, 686, 507, 513, 351, 243, 405, 7467, 6419, 5575, 735, 6859, 1813, 3231, 1183, 1197, 3537, 819, 1647, 567, 945, 2197, 8397, 3211, 1715, 3249, 3367

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355737.

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.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     6: {1,2}
     9: {2,2}
    15: {2,3}
    49: {4,4}
    35: {3,4}
    27: {2,2,2}
    45: {2,2,3}
    98: {1,4,4}
    63: {2,2,4}
   105: {2,3,4}
   171: {2,2,8}
   117: {2,2,6}
    81: {2,2,2,2}
   135: {2,2,2,3}
For example, the choices for a(12) = 105 are:
  (1,1,1)  (1,3,2)  (2,1,4)
  (1,1,2)  (1,3,4)  (2,3,1)
  (1,1,4)  (2,1,1)  (2,3,2)
  (1,3,1)  (2,1,2)  (2,3,4)

Wikipedia, Coprime integers.

Not requiring coprimality gives A355732, firsts of A355731.
Positions of first appearances in A355737.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A007359, A051424, A076610, A302696, A302698, A355733, A355735, A355739, A355741, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}];
Table[Position[az+1,k][[1,1]],{k,mnrm[az+1]}]

A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 2, 4, 12, 24, 92, 184, 704, 2016, 7600, 15200, 80664, 161328, 601696, 2198824, 9868544, 19737088, 102010480, 204020960

Offset: 0

Gus Wiseman, Mar 13 2025

A constant partition is a multiset whose parts are all equal. There are A000005(n) constant partitions of n.

			The a(1) = 1 through a(4) = 12 multisets:
  {1}  {1,2}    {1,2,3}        {1,2,3,4}
       {1,1,1}  {1,1,1,3}      {1,1,1,3,4}
                {1,1,1,1,2}    {1,2,2,2,3}
                {1,1,1,1,1,1}  {1,1,1,1,2,4}
                               {1,1,1,2,2,3}
                               {1,1,1,1,1,1,4}
                               {1,1,1,1,1,2,3}
                               {1,1,1,1,2,2,2}
                               {1,1,1,1,1,1,1,3}
                               {1,1,1,1,1,1,2,2}
                               {1,1,1,1,1,1,1,1,2}
                               {1,1,1,1,1,1,1,1,1,1}

The number of possible choices was A066843.
Multiset partitions into constant blocks: A006171, A279784, A295935.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Sets of constant multisets with distinct sums: A381635, A381636, A381716.
Strict instead of constant partitions: A381808, A058694, A152827.
A000041 counts integer partitions, strict A000009, constant A000005.
A000688 counts multiset partitions into constant blocks.
A050361 and A381715 count multiset partitions into constant multisets.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@Range[n]]]],{n,0,10}]

Primorial case of A381453: a(n) = A381453(A002110(n)).

a(16)-a(19) from Christian Sievers, Jun 04 2025

A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 38, 145, 586, 2619, 12096, 58370, 285244, 1436815, 7281062, 37489525, 193417612

Offset: 0

Gus Wiseman, Mar 14 2025

			The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,2,3}    {1,2,3,4}      {1,2,3,4,5}
              {1,1,2,2}  {1,1,2,2,4}    {1,1,2,2,4,5}
                         {1,1,2,3,3}    {1,1,2,3,3,5}
                         {1,1,1,2,2,3}  {1,1,2,3,4,4}
                                        {1,2,2,3,3,4}
                                        {1,1,1,2,2,3,5}
                                        {1,1,1,2,2,4,4}
                                        {1,1,1,2,3,3,4}
                                        {1,1,2,2,2,3,4}
                                        {1,1,2,2,3,3,3}
                                        {1,1,1,1,2,2,3,4}
                                        {1,1,1,2,2,2,3,3}

Set systems: A050342, A116539, A296120, A318361.
The number of possible choices was A152827, non-strict A058694.
Set multipartitions with distinct sums: A279785, A381718.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Constant instead of strict partitions: A381807, A066843.
A000041 counts integer partitions, strict A000009, constant A000005.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@Range[n]]]],{n,0,10}]

a(12)-a(16) from Christian Sievers, Jun 04 2025

A387114 Number of divisors in common to all prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 2, 1, 4, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 4, 1, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Aug 19 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.

Also the number of divisors of the greatest common divisor of the prime indices of n.

			The prime indices of 703 are {8,12}, with divisors {{1,2,4,8},{1,2,3,4,6,12}}, with {1,2,4} in common, so a(703) = 3.

For initial interval instead of divisors we have A055396.
Positions of 1 are A289509, complement A318978.
Positions of 2 are A387119.
For prime factors or indices instead of divisors we have A387135, see A010055 or A069513.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives greatest common divisor of prime indices.
Cf. A000720, A061395, A326841, A335433, A335448, A355731, A355733, A355734, A355737, A355739, A355740, A370820.

Table[If[n==1,0,Length[Divisors[GCD@@PrimePi/@First/@FactorInteger[n]]]],{n,100}]

a(1) = 0; a(n) = A000005(A289508(n)) for n > 1.

A387119 Numbers whose prime indices all have exactly 2 divisors in common.

Original entry on oeis.org

3, 5, 9, 11, 17, 21, 25, 27, 31, 39, 41, 57, 59, 63, 65, 67, 81, 83, 87, 91, 109, 111, 115, 117, 121, 125, 127, 129, 147, 157, 159, 171, 179, 183, 185, 189, 191, 203, 211, 213, 235, 237, 241, 243, 247, 261, 267, 273, 277, 283, 289, 299, 301, 303, 305, 319, 321

Offset: 1

Gus Wiseman, Aug 21 2025

All terms are odd.

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.

			The prime indices of 87 are {2,10}, with divisors {{1,2},{1,2,5,10}}, with intersection {1,2}, so 87 is in the sequence.
The prime indices of 91 are {4,6}, with divisors {{1,2,4},{1,2,3,6}}, with intersection {1,2}, so 91 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    5: {3}
    9: {2,2}
   11: {5}
   17: {7}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   31: {11}
   39: {2,6}
   41: {13}
   57: {2,8}
   59: {17}
   63: {2,2,4}
   65: {3,6}
   67: {19}
   81: {2,2,2,2}

For initial intervals instead of divisors we have A016945.
Positions of 1 are A289509, complement A318978.
Positions of 2 in A387114, for prime factors or indices A387135.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives greatest common divisor of prime indices.
Cf. A000720, A055396, A061395, A335433, A355731, A355733, A355739, A355740, A370820.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Length[Intersection@@Divisors/@prix[#]]==2&]

cp's OEIS Frontend

A355537 Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

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A355748 Number of ways to choose a sequence of divisors, one of each part of the n-th composition in standard order.

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A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

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A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

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A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

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A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

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A387114 Number of divisors in common to all prime indices of n.

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A387119 Numbers whose prime indices all have exactly 2 divisors in common.

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