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.

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A367915 Sorted positions of first appearances in A367912 (number of multisets that can be obtained by choosing a binary index of each binary index).

Original entry on oeis.org

1, 4, 20, 52, 64, 68, 84, 116, 308, 320, 324, 340, 372, 816, 832, 836, 848, 852, 880, 884, 1104, 1108, 1136, 1360, 1364, 1392, 1396, 1904, 1908, 2868, 2884, 2900, 2932, 3152, 3184, 3188, 3412, 3424, 3440, 3444, 3952, 3956, 5188, 5204, 5216, 5220, 5236, 5476
Offset: 1

Views

Author

Gus Wiseman, Dec 16 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The terms together with the corresponding set-systems begin:
     1: {{1}}
     4: {{1,2}}
    20: {{1,2},{1,3}}
    52: {{1,2},{1,3},{2,3}}
    64: {{1,2,3}}
    68: {{1,2},{1,2,3}}
    84: {{1,2},{1,3},{1,2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   308: {{1,2},{1,3},{2,3},{1,4}}
   320: {{1,2,3},{1,4}}
   324: {{1,2},{1,2,3},{1,4}}
   340: {{1,2},{1,3},{1,2,3},{1,4}}
   372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

Crossrefs

A version for multisets and divisors is A355734.
Sorted positions of first appearances in A367912, for sequences A368109.
The unsorted version is A367913.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355744, A367905, A367906, A367911, A368112, A368185.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,10000}];
Select[Range[Length[c]],FreeQ[Take[c,#-1],c[[#]]]&]

A387115 Number of ways to choose a sequence of distinct strict integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 0, 0, 2, 3, 0, 4, 2, 2, 0, 5, 0, 6, 0, 2, 3, 8, 0, 2, 4, 0, 0, 10, 2, 12, 0, 3, 5, 4, 0, 15, 6, 4, 0, 18, 2, 22, 0, 0, 8, 27, 0, 2, 2, 5, 0, 32, 0, 6, 0, 6, 10, 38, 0, 46, 12, 0, 0, 8, 3, 54, 0, 8, 4, 64, 0, 76, 15, 2, 0, 6, 4, 89, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Aug 20 2025

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.
The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

Examples

			The prime indices of 15 are (2,3), and there are a(15) = 2 choices:
  ((2),(3))
  ((2),(2,1))
The prime indices of 121 are (5,5), and there are a(121) = 6 choices:
  ((5),(4,1))
  ((5),(3,2))
  ((4,1),(5))
  ((4,1),(3,2))
  ((3,2),(5))
  ((3,2),(4,1))

Crossrefs

For divisors instead of partitions we have A355739, see A355740, A355733, A355734.
Allowing repeated partitions gives A357982, see A299200, A357977, A357978.
Twice-partitions of this type are counted by A358914, strict case of A270995.
The disjoint case is A383706.
Allowing non-strict partitions gives A387110, for prime factors A387133.
For initial intervals instead of strict partitions we have A387111.
For constant instead of strict partitions we have A387120.
Positions of 0 are A387176 (non-choosable), complement A387177 (choosable).
A000041 counts integer partitions, strict A000009.
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.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A063834, A261049, A335433, A335448, A355731, A355741, A355742, A355744, A357980.

Programs

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

A355736 Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.

Original entry on oeis.org

1, 3, 7, 13, 21, 37, 39, 89, 133, 117, 111, 273, 351, 259, 267, 333, 453, 793, 669, 623, 999, 777, 843, 1491, 1157, 1561, 2863, 1443, 1963, 2331, 1977, 1869, 2899, 2529, 3207, 4107, 3171, 5073, 4329, 3653, 4667, 3471, 7399, 4613, 7587, 5931, 7269, 5889, 7483
Offset: 1

Views

Author

Gus Wiseman, Jul 21 2022

Keywords

Comments

This is the position of first appearance of n in A355735.
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:
     1: {}
     3: {2}
     7: {4}
    13: {6}
    21: {2,4}
    37: {12}
    39: {2,6}
    89: {24}
   133: {4,8}
   117: {2,2,6}
   111: {2,12}
   273: {2,4,6}
   351: {2,2,2,6}
For example, the choices for a(12) = 273 are:
  {1,1,1}  {1,2,2}  {2,2,2}
  {1,1,2}  {1,2,3}  {2,2,3}
  {1,1,3}  {1,2,6}  {2,2,6}
  {1,1,6}  {1,4,6}  {2,4,6}

Crossrefs

Allowing any choice of divisors gives A355732, firsts of A355731.
Choosing a multiset instead of sequence gives A355734, firsts of A355733.
Positions of first appearances in A355735.
The case of prime factors instead of divisors is counted by A355745.
The decreasing version is counted by A355749.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A355737, A355739, A355740, A355741, A355744.

Programs

  • Mathematica
    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]],LessEqual@@#&]],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]

A355749 Number of ways to choose a weakly decreasing sequence of divisors, one of each prime index of n (with multiplicity, taken in weakly increasing order).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 3, 1, 3, 1, 3, 1, 4, 1, 4, 1, 2, 1, 2, 1, 3, 1, 6, 1, 3, 1, 2, 1, 4, 1, 3, 1, 4, 1, 6, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 6, 1, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 1, 2, 1, 3
Offset: 1

Views

Author

Gus Wiseman, Jul 18 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 a(2) = 1 through a(19) = 4 choices:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
     2      3      2       21      5       2      21        7       2
                   4       22              3                        4
                                           6                        8

Links

Crossrefs

Allowing any choice of divisors gives A355731, firsts A355732.
Choosing a multiset instead of sequence gives A355733, firsts A355734.
The reverse version is A355735, firsts A355736, only primes A355745.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061395 selects the maximum prime index.
Cf. A000720, A076610, A120383, A316524, A324850, A355737, A355739, A355740, A355741, A355742, A355744.

Programs

  • Mathematica
    Table[Length[Select[Tuples[Divisors/@primeMS[n]], GreaterEqual@@#&]],{n,100}]

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

Views

Author

Gus Wiseman, Aug 19 2025

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

Examples

			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.

Crossrefs

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.

Programs

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

Formula

a(1) = 0; a(n) = A000005(A289508(n)) for n > 1.
Previous Showing 11-15 of 15 results.

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