cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.

Original entry on oeis.org

9, 21, 25, 27, 45, 49, 57, 63, 75, 81, 99, 105, 115, 117, 121, 125, 133, 135, 147, 153, 159, 171, 175, 189, 195, 207, 225, 231, 243, 245, 261, 273, 275, 279, 285, 289, 297, 315, 325, 333, 343, 345, 351, 357, 361, 363, 369, 371, 375, 387, 393, 399, 405, 423
Offset: 1

Views

Author

Gus Wiseman, Jul 22 2022

Keywords

Comments

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.

Examples

			The terms together with their prime indices begin:
    9: {2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   45: {2,2,3}
   49: {4,4}
   57: {2,8}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   99: {2,2,5}
  105: {2,3,4}
For example, the prime indices of 897 are {2,6,9}, of which we can choose prime factors in two ways: (2,2,3) or (2,3,3); but neither of these has all distinct elements, so 897 is in the sequence.

Crossrefs

Including evens gives A355529.
The version for all divisors including evens is A355740, zeros of A355739.
Choices of a prime factor of each prime index: A355741, unordered A355744.
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.
Cf. A000720, A076610, A289509, A302796, A327486, A355731, A355733, A355742.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&Select[Tuples[primeMS/@primeMS[#]],UnsameQ@@#&]=={}&]

A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 58, 116, 320, 772, 2170, 4340, 14112, 28224, 78120, 212004, 612232, 1224464, 3873760, 7747520, 24224608, 64595088, 175452168, 350904336
Offset: 0

Views

Author

Gus Wiseman, Jul 20 2022

Keywords

Examples

			The a(0) = 1 through a(4) = 10 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
           {1,2}  {1,1,2}  {1,1,1,2}
                  {1,1,3}  {1,1,1,3}
                  {1,2,3}  {1,1,1,4}
                           {1,1,2,2}
                           {1,1,2,3}
                           {1,1,2,4}
                           {1,1,3,4}
                           {1,2,2,3}
                           {1,2,3,4}

Crossrefs

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
Counting sequences instead of multisets gives A066843.
The integers themselves are the rows of A131818 (shifted).
For prime indices we have A355733, only prime factors A355744.
For prime factors instead of divisors we have A355746, factors A355537.
A000005 counts divisors.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
Cf. A000720, A002110, A006218, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

Programs

  • Mathematica
    Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]],{n,0,10}]
  • Python
    from sympy import divisors
from itertools import count, islice
def agen():
    s = {tuple()}
    for n in count(1):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in divisors(n))
print(list(islice(agen(), 16))) # Michael S. Branicky, Aug 03 2022

Formula

a(n) = A355733(A070826(n)).
a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

Extensions

a(15)-a(21) from Michael S. Branicky, Aug 03 2022
a(22)-a(23) from Michael S. Branicky, Aug 08 2022

A355746 Number of different multisets that can be obtained by choosing a prime index (or a prime factor) of each integer from 2 to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 12, 20, 20, 20, 26, 26, 36, 58, 116, 116, 140, 140, 280, 280, 384, 384, 536, 536, 536, 844, 1688, 2380, 2716, 2716, 5432, 8484, 10152, 10152, 13308, 13308, 18064, 21616, 43232, 43232, 47648, 47648, 54656, 84480, 114304, 114304
Offset: 1

Views

Author

Gus Wiseman, Jul 20 2022

Keywords

Examples

			The a(n) multisets for n = 2, 6, 10, 12:
  {1}  {1,1,1,2,3}  {1,1,1,1,1,2,2,3,4}  {1,1,1,1,1,1,2,2,3,4,5}
       {1,1,2,2,3}  {1,1,1,1,2,2,2,3,4}  {1,1,1,1,1,2,2,2,3,4,5}
                    {1,1,1,1,2,2,3,3,4}  {1,1,1,1,1,2,2,3,3,4,5}
                    {1,1,1,2,2,2,3,3,4}  {1,1,1,1,2,2,2,2,3,4,5}
                                         {1,1,1,1,2,2,2,3,3,4,5}
                                         {1,1,1,2,2,2,2,3,3,4,5}

Links

Crossrefs

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
The integers themselves are the rows of A131818 (shifted).
Counting sequences instead of multisets: A355537, with multiplicity A327486.
Using prime indices instead of 2..n gives A355744, for sequences A355741.
The version for divisors instead of prime factors is A355747.
A000040 lists the prime numbers.
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. A000720, A002110, A076610, A355538, A355731, A355733, A355740, A355742, A355745.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Tuples[primeMS/@Range[2,n]]]],{n,15}]
  • Python
    from sympy import factorint
from itertools import count, islice
def agen():
    s = {(1,)}
    for n in count(2):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in factorint(n))
print(list(islice(agen(), 53))) # Michael S. Branicky, Aug 03 2022

Formula

a(n) = A355744(A070826(n)).
a(p) = a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

Extensions

a(28) and beyond from Michael S. Branicky, Aug 03 2022

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 20 2022

Keywords

Comments

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.

Examples

			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)

Crossrefs

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.

Programs

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

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

Views

Author

Gus Wiseman, Jul 23 2022

Keywords

Comments

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

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Gus Wiseman, Mar 13 2025

Keywords

Comments

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

Examples

			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}

Crossrefs

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.

Programs

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

Formula

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

Extensions

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

Views

Author

Gus Wiseman, Mar 14 2025

Keywords

Examples

			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}

Crossrefs

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.

Programs

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

Extensions

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

A327485 Product of means of integer partitions with Heinz numbers from 2 to n.

Original entry on oeis.org

1, 2, 2, 6, 9, 36, 36, 72, 144, 720, 960, 5760, 14400, 36000, 36000, 252000, 420000, 3360000, 5600000, 16800000, 50400000, 453600000, 567000000, 1701000000, 5953500000, 11907000000, 23814000000, 238140000000, 476280000000, 5239080000000, 5239080000000
Offset: 2

Views

Author

Gus Wiseman, Sep 28 2019

Keywords

Comments

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.

Examples

			We have a(7) = 1 * 2 * 1 * 3 * 3/2 * 4 = 36.

Crossrefs

Partial products of A326567/A326568.
Cf. A056239, A112798, A316413, A327474, A327482, A327486.

Programs

  • Mathematica
    Table[Product[Total[Cases[FactorInteger[k],{p_,k_}:>k*PrimePi[p]]]/PrimeOmega[k],{k,2,n}],{n,2,30}]

Formula

a(n > 2) = a(n - 1) * A326567(n) / A326568(n).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.