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 1-3 of 3 results.

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}]
Showing 1-3 of 3 results.

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