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.

Previous Showing 21-28 of 28 results.

A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 39, 41, 51, 55, 59, 65, 67, 83, 85, 87, 93, 109, 111, 123, 127, 129, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 213, 235, 237, 241, 249, 255, 267, 277, 283, 295, 303, 305, 319, 321, 327, 331, 335, 341, 353, 365, 367, 381
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2023

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 prime indices of 2795 are {3,6,14}, with prime factors {{3},{2,3},{2,7}}, and the only choice with different terms is {3,2,7}, so 2795 is in the sequence.
The terms together with their prime indices of prime indices begin:
    1: {}
    3: {{1}}
    5: {{2}}
   11: {{3}}
   15: {{1},{2}}
   17: {{4}}
   31: {{5}}
   33: {{1},{3}}
   39: {{1},{1,2}}
   41: {{6}}
   51: {{1},{4}}
   55: {{2},{3}}
   59: {{7}}
   65: {{2},{1,2}}
   67: {{8}}
   83: {{9}}
   85: {{2},{4}}
   87: {{1},{1,3}}
   93: {{1},{5}}
  109: {{10}}
  111: {{1},{1,1,2}}

Crossrefs

For no choices we have A355529, odd A355535, binary A367907.
Positions of ones in A367771.
The version for binary indices is A367908, positions of ones in A367905.
For any number of choices we have A368100.
For a unique set instead of sequence we have A370647, counted by A370594.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sort A118914, length A001221, sum A001222.
A355741 chooses a prime factor of each prime index, multisets A355744.
Cf. A007716, A355737, A355739, A355740, A355745, A367904, A367906, A370584.

Programs

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

A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

0, 1, 2, 3, 7, 10, 15, 22, 61, 81, 112, 154, 207, 276, 355, 464, 1771, 2166, 2724, 3445, 4246, 5292, 6420, 7922, 9586, 11667, 13768, 16606, 19095, 22825, 26498, 31421, 187223, 213684, 247670, 289181, 331301, 385079, 440411, 510124, 575266, 662625, 747521
Offset: 0

Views

Author

Gus Wiseman, Mar 08 2024

Keywords

Comments

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.

Examples

			The a(0) = 0 through a(6) = 15 subsets:
  .  {1}  {2}    {3}    {4}      {5}      {6}
          {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
                 {2,3}  {2,4}    {2,5}    {2,6}
                        {3,4}    {3,5}    {3,6}
                        {1,2,4}  {4,5}    {4,6}
                        {1,3,4}  {1,2,5}  {5,6}
                        {2,3,4}  {1,3,5}  {1,2,6}
                                 {2,3,5}  {1,3,6}
                                 {2,4,5}  {1,4,6}
                                 {3,4,5}  {1,5,6}
                                          {2,3,6}
                                          {2,5,6}
                                          {3,4,6}
                                          {3,5,6}
                                          {4,5,6}

Links

Crossrefs

Simple graphs of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A134964, complement A140637.
Simple graphs not of this type are counted by A367867, covering A367868.
Set systems of this type are counted by A367902, ranks A367906.
Set systems not of this type are counted by A367903, ranks A367907.
Set systems uniquely of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
Factorizations of this type are A368414/A370814, complement A368413/A370813.
For prime instead of binary indices we have A370586, differences of A370582.
The complement for prime indices is A370587, differences of A370583.
The complement is counted by A370589, differences of A370637.
Partial sums are A370636.
The complement has partial sums A370637/A370643, minima A370642/A370644.
The case of a unique choice is A370641, differences of A370638.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A326702, A355739, A355740, A367770, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]

Formula

First differences of A370636.

Extensions

a(19)-a(42) from Alois P. Heinz, Mar 09 2024

A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 1, 6, 17, 42, 67, 175, 400, 870, 1841, 3820, 7837, 15920, 30997, 63370, 128348, 258699, 520042, 1043284, 2090732, 4186382, 8379022, 16765549, 33540664, 67092258, 134198633, 268412631, 536844414, 1073710403, 2147296425, 4294753612, 8589686922, 17179580003
Offset: 0

Views

Author

Gus Wiseman, Mar 08 2024

Keywords

Comments

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.

Examples

			The binary indices of {1,4,5} are {{1},{3},{1,3}}, from which it is not possible to choose three different elements, so S is counted under a(3).
The binary indices of S = {1,6,8,9} are {{1},{2,3},{4},{1,4}}, from which it is not possible to choose four different elements, so S is counted under a(9).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3,4}  {1,4,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {3,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

Links

Crossrefs

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370586, differences of A370582.
For prime indices we have A370587, differences of A370583.
Partial sums are A370637/A370643, minima A370642/A370644.
The complement is counted by A370639, partial sums A370636.
The version for a unique choice is A370641, partial sums A370638.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A072639, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

Extensions

a(19)-a(35) from Alois P. Heinz, Mar 09 2024

A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Mar 01 2024

Keywords

Comments

All of these factorizations are co-balanced (A340596).

Examples

			The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
  (2)  (2*6)  (3*8)   (4*9)   (8*9)   (3*5*8)   (2*72)   (4*5*9)   (3*96)
       (3*4)  (4*6)   (6*6)   (2*36)  (4*5*6)   (3*48)   (5*6*6)   (4*72)
              (2*12)  (2*18)  (3*24)  (2*3*20)  (4*36)   (2*3*30)  (6*48)
                      (3*12)  (4*18)  (2*5*12)  (6*24)   (2*5*18)  (8*36)
                              (6*12)  (2*6*10)  (8*18)   (2*6*15)  (9*32)
                                      (3*4*10)  (9*16)   (2*9*10)  (12*24)
                                                (12*12)  (3*4*15)  (16*18)
                                                         (3*5*12)  (2*144)
                                                         (3*6*10)

Crossrefs

Multisets of this type are ranked by A368101, see also A368100, A355529.
For nonexistence we have A368413, complement A368414.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, see also A370592, A370593.
Subsets of this type are counted by A370638, see also A370636, A370637.
For unlabeled multiset partitions we have A370646, also A368098, A368097.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A027746 lists prime factors, A112798 indices, length A001222.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
A355741 counts ways to choose a prime factor of each prime index.
For set-systems see A367902-A367908.
Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]

A370647 Numbers such that only one set can be obtained by choosing a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 23, 31, 33, 35, 39, 41, 51, 53, 55, 59, 65, 67, 69, 77, 83, 85, 87, 91, 93, 95, 97, 103, 109, 111, 119, 123, 127, 129, 131, 155, 157, 161, 165, 169, 177, 179, 183, 185, 187, 191, 201, 203, 205, 209, 211, 213, 217, 227, 235, 237, 241
Offset: 1

Views

Author

Gus Wiseman, Mar 06 2024

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 prime indices of 91 are {4,6}, with only choice {2,3}, so 91 is in the sequence.
The terms together with their prime indices begin:
     1: {}        53: {16}      109: {29}
     3: {2}       55: {3,5}     111: {2,12}
     5: {3}       59: {17}      119: {4,7}
     7: {4}       65: {3,6}     123: {2,13}
    11: {5}       67: {19}      127: {31}
    15: {2,3}     69: {2,9}     129: {2,14}
    17: {7}       77: {4,5}     131: {32}
    19: {8}       83: {23}      155: {3,11}
    23: {9}       85: {3,7}     157: {37}
    31: {11}      87: {2,10}    161: {4,9}
    33: {2,5}     91: {4,6}     165: {2,3,5}
    35: {3,4}     93: {2,11}    169: {6,6}
    39: {2,6}     95: {3,8}     177: {2,17}
    41: {13}      97: {25}      179: {41}
    51: {2,7}    103: {27}      183: {2,18}

Crossrefs

For nonexistence we have A355529, count A370593.
For binary instead of prime indices we have A367908, counted by A367904.
For existence we have A368100, count A370592.
For a sequence instead of set of factors we have A368101.
The version for subsets is A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594.
For subsets and binary indices we have A370638.
The version for factorizations is A370645, see also A368414, A368413.
For divisors instead of factors we have A370810, counted by A370595.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000720, A003963, A355739, A355740, A355744, A355745, A368110.

Programs

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

A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 15, 9, 1, 0, 316, 198, 28, 1, 0, 16885, 10710, 1610, 75, 1, 0, 2174586, 1384335, 211820, 10575, 186, 1, 0, 654313415, 416990763, 64144675, 3268125, 61845, 441, 1, 0, 450179768312, 286992935964, 44218682312, 2266772550, 43832264, 336924, 1016, 1
Offset: 0

Views

Author

Geoffrey Critzer, Apr 02 2023

Keywords

Comments

Also the number of sets of n nonempty subsets of {1..n}, k of which are singletons, such that there is only one way to choose a different element from each. For example, row n = 3 counts the following set-systems:
{{1},{1,2},{1,3}} {{1},{2},{1,3}} {{1},{2},{3}}
{{1},{1,2},{2,3}} {{1},{2},{2,3}}
{{1},{1,3},{2,3}} {{1},{3},{1,2}}
{{2},{1,2},{1,3}} {{1},{3},{2,3}}
{{2},{1,2},{2,3}} {{2},{3},{1,2}}
{{2},{1,3},{2,3}} {{2},{3},{1,3}}
{{3},{1,2},{1,3}} {{1},{2},{1,2,3}}
{{3},{1,2},{2,3}} {{1},{3},{1,2,3}}
{{3},{1,3},{2,3}} {{2},{3},{1,2,3}}
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{1,2,3}}
{{3},{2,3},{1,2,3}}

Examples

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,    15,     9,    1;
  0,   316,   198,   28,  1;
  0, 16885, 10710, 1610, 75, 1;
  ...

Links

Crossrefs

Cf. A058876 (mirror), A361579, A224069.
Row-sums are A003024, unlabeled A003087.
Column k = 1 is A003025(n) = |n*A134531(n)|.
Column k = n-1 is A058877.
For fixed sinks we get A368602.
A058891 counts set-systems, unlabeled A000612.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A059201, A082402, A088957, A133686, A334282, A350415, A367904, A367908, A368600, A368601.

Programs

  • Mathematica
    nn = 8; B[n_] := n! 2^Binomial[n, 2] ;ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /. Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[ Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-z]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid
nv=4;Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]],{n,0,nv},{k,0,n}]

Formula

T(n,k) = A368602(n,k) * binomial(n,k). - Gus Wiseman, Jan 03 2024

A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 79, 33, 7, 1, 0, 3377, 1071, 161, 15, 1, 0, 362431, 92289, 10591, 705, 31, 1, 0, 93473345, 19856703, 1832705, 93375, 2945, 63, 1, 0, 56272471039, 10249747713, 789619327, 32382465, 782719, 12033, 127, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 02 2024

Keywords

Comments

Also the number of set-systems with n vertices and n edges such that {i} is a singleton edge iff i <= k, and such that there is only one way to choose a different vertex from each edge.

Examples

			Triangle begins:
    1
    0    1
    0    1    1
    0    5    3    1
    0   79   33    7    1
    0 3377 1071  161   15    1
    ...
Row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{2},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}

Crossrefs

Column k = n-1 is A000225 = A058877(n)/n.
Column k = 1 is A134531 (up to sign) or A003025(n)/n, non-fixed A350415.
For any choice of k sinks we get A361718.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A003024, A003087, A082402, A088957, A334282, A367862, A367904, A367908, A368600, A368601.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Union@@Cases[#,{_}]==Range[k] && Length[Select[Tuples[#],UnsameQ@@#&]]==1&]], {n,0,3},{k,0,n}]

Formula

T(n,k) = A361718(n,k)/binomial(n,k).

Extensions

More terms from Alois P. Heinz, Jan 04 2024

A370818 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.

Original entry on oeis.org

1, 2, 6, 45, 1352, 157647, 63380093, 85147722812, 385321270991130
Offset: 0

Views

Author

Gus Wiseman, Mar 12 2024

Keywords

Examples

			The set-system {{2},{1,2},{2,4},{1,3,4}} has unique choice (2,1,4,3) so is counted under a(4).

Crossrefs

This is the unique version of A367902, complement A367903.
Choosing a sequence gives A367904, ranks A367908.
The maximal case is A368601, complement A368600.
This is the restriction of A370638 to A000225.
Factorizations of this type are counted by A370645.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A133686, A134964, A367772, A367867, A368101, A368109, A370584.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Union[Sort/@Select[Tuples[#],UnsameQ@@#&]]]==1&]],{n,0,3}]

Formula

a(n) = A370638(2^n - 1).
Binomial transform of A368601. - Christian Sievers, Aug 12 2024

Extensions

a(5)-a(8) from Christian Sievers, Aug 12 2024
Previous Showing 21-28 of 28 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.