A355740 -id:A355740 - cp's OEIS frontend

A140638 Number of connected graphs on n labeled nodes that contain at least two cycles.

0, 0, 0, 7, 381, 21748, 1781154, 249849880, 66257728763, 34495508486976, 35641629989151608, 73354595357480683904, 301272202621204113362497, 2471648811029413368450098688, 40527680937730440155535277704046, 1328578958335783199341353852258282496

Offset: 1

Washington Bomfim, May 21 2008

nonn

These are the connected graphs that are neither trees nor unicyclic.

Also connected non-choosable graphs covering n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. The unlabeled version is A140636. The complement is counted by A129271. - Gus Wiseman, Feb 20 2024

J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.

Andrew Howroyd, Table of n, a(n) for n = 1..50

The unlabeled version is A140636.
Cf. A000272 (trees), A001187 (connected graphs), A057500 (connected unicyclic graphs).
The complement is counted by A129271, unlabeled A005703.
The non-connected complement is A133686, covering A367869.
The non-connected version is A367867, unlabeled A140637.
The non-connected covering version is A367868.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A143543 counts simple labeled graphs by number of connected components.
Cf. A001349, A116508, A355740, A367769, A367863, A367901, A367903, A370317.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,5}] (* Gus Wiseman, Feb 19 2024 *)

seq(n)={my(A=O(x*x^n), t=-lambertw(-x + A)); Vec(serlaplace( log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, A)) - log(1/(1-t))/2 - t/2 + 3*t^2/4), -n)} \\ Andrew Howroyd, Jan 15 2022

a(n) = A001187(n) - A129271(n).

a(n) = A001187(n) - A000272(n) - A057500(n).

Definition clarified by Andrew Howroyd, Jan 15 2022

A355737 Number of ways to choose a sequence of divisors, one of each prime index of n (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 4, 1, 1, 4, 1, 2, 4, 2, 1, 2, 3, 4, 7, 3, 1, 4, 1, 1, 4, 2, 6, 4, 1, 4, 6, 2, 1, 6, 1, 2, 8, 3, 1, 2, 5, 4, 4, 4, 1, 8, 4, 3, 5, 4, 1, 4, 1, 2, 10, 1, 6, 4, 1, 2, 6, 6, 1, 4, 1, 6, 8, 4, 6, 8, 1, 2, 15, 2, 1, 6, 4, 4

Offset: 1

Gus Wiseman, Jul 17 2022

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.

			The a(2) = 1 through a(18) = 4 choices:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111
               12          12  13     112     12  13           112
                           21                 14  21           121
                                                  23           122

Wikipedia, Coprime integers.

Dominated by A355731, firsts A355732, primes A355741, prime-powers A355742.
For weakly increasing instead of coprime we have A355735, primes A355745.
Positions of first appearances are A355738.
For strict instead of coprime we have A355739, zeros A355740.
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, A289507, A296150, A302696, A302698, A355733, A355744, A355748.

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

A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 10, 11, 14, 17, 19, 23, 29, 30, 39, 41, 51, 58, 66, 78, 82, 102, 110, 132, 144, 162, 186, 210, 228, 260, 296, 328, 366, 412, 462, 512, 560, 638, 692, 764, 860, 924, 1028, 1122, 1276, 1406, 1528, 1721, 1898, 2056, 2318, 2506, 2812, 3020, 3442

Offset: 0

Gus Wiseman, Mar 05 2024

nonn

			For the partitions of 5 we have the following choices:
      (5): {{1},{5}}
     (41): {{1,1},{1,2},{1,4}}
     (32): {{1,1},{1,2},{1,3},{2,3}}
    (311): {{1,1,1},{1,1,3}}
    (221): {{1,1,1},{1,1,2},{1,2,2}}
   (2111): {{1,1,1,1},{1,1,1,2}}
  (11111): {{1,1,1,1,1}}
So a(5) = 4.

For just prime factors we have A370809.
The version for factorizations is A370816, for just prime factors A370817.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts condensed partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355733 counts choices of divisors of prime indicec.
A370320 counts non-condensed partitions, ranks A355740.
A370592 counts factor-choosable partitions, complement A370593.
Cf. A000792, A048249, A066739, A319055, A355737, A355739, A370348, A370595, A370803, A370810.

Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@IntegerPartitions[n]],{n,0,30}]

Terms a(31) onward from Max Alekseyev, Sep 17 2024

A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 04 2024

nonn

A multiset is condensed iff it is possible to choose a different divisor of each element.

			The a(96) = 4 factorizations: (2*2*2*2*2*3), (2*2*2*2*6), (2*2*2*3*4), (2*2*2*12).

Partitions not of this type are counted by A239312, ranks A368110.
Factors instead of divisors: A368413, complement A368414, unique A370645.
Partitions of this type are counted by A370320, ranks A355740.
Subsets of this type: A370583 and A370637, complement A370582 and A370636.
The complement is counted by A370814, partitions A370592, ranks A368100.
For a unique choice we have A370815, partitions A370595, ranks A370810.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A340596, A340653, A355529, A355739, A355741, A370804, A370807.

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[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]==0&]],{n,100}]

A370320 Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 9, 13, 20, 28, 40, 54, 74, 102, 135, 180, 235, 310, 397, 516, 658, 843, 1066, 1349, 1687, 2119, 2634, 3273, 4045, 4995, 6128, 7517, 9171, 11181, 13579, 16457, 19884, 23992, 28859, 34646, 41506, 49634, 59211, 70533, 83836, 99504, 117867

Offset: 0

Gus Wiseman, Mar 02 2024

nonn

Includes all partitions containing 1.

			The a(0) = 0 through a(8) = 13 partitions:
  .  .  (11)  (111)  (211)   (221)    (222)     (331)      (611)
                     (1111)  (311)    (411)     (511)      (2222)
                             (2111)   (2211)    (2221)     (3221)
                             (11111)  (3111)    (3211)     (3311)
                                      (21111)   (4111)     (4211)
                                      (111111)  (22111)    (5111)
                                                (31111)    (22211)
                                                (211111)   (32111)
                                                (1111111)  (41111)
                                                           (221111)
                                                           (311111)
                                                           (2111111)
                                                           (11111111)

The complement is counted by A239312 (condensed partitions).
These partitions have ranks A355740.
Factorizations in the case of prime factors are A368413, complement A368414.
The complement for prime factors is A370592, ranks A368100.
The version for prime factors (not all divisors) is A370593, ranks A355529.
For a unique choice we have A370595, ranks A370810.
For multiple choices we have A370803, ranks A370811.
The case without ones is A370804, complement A370805.
The version for factorizations is A370813, complement A370814.
A000005 counts divisors.
A000041 counts integer partitions.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A355535, A355739, A367867, A368097, A368110, A370583, A370584, A370594, A370806, A370807, A370808.

Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]==0&]],{n,0,30}]

a(31)-a(47) from Alois P. Heinz, Mar 03 2024

A370814 Number of condensed integer factorizations of n into unordered factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Mar 04 2024

nonn

A multiset is condensed iff it is possible to choose a different divisor of each element.

			The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).

Partitions of this type are counted by A239312, ranks A368110.
Factors instead of divisors: A368414, complement A368413, unique A370645.
Partitions not of this type are counted by A370320, ranks A355740.
Subsets of this type: A370582 and A370636, complement A370583 and A370637.
The complement is counted by A370813, partitions A370593, ranks A355529.
For a unique choice we have A370815, partitions A370595, ranks A370810.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A340596, A340653, A355739, A370592, A370803, A370804, A370805.

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[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]>0&]],{n,100}]

A370583 Number of subsets of {1..n} such that it is not possible to choose a different prime factor of each element.

Original entry on oeis.org

0, 1, 2, 4, 10, 20, 44, 88, 204, 440, 908, 1816, 3776, 7552, 15364, 31240, 63744, 127488, 257592, 515184, 1036336, 2079312, 4166408, 8332816, 16709632, 33470464, 66978208, 134067488, 268236928, 536473856, 1073233840, 2146467680, 4293851680, 8588355424, 17177430640

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

			The a(0) = 0 through a(5) = 20 subsets:
  .  {1}  {1}    {1}      {1}        {1}
          {1,2}  {1,2}    {1,2}      {1,2}
                 {1,3}    {1,3}      {1,3}
                 {1,2,3}  {1,4}      {1,4}
                          {2,4}      {1,5}
                          {1,2,3}    {2,4}
                          {1,2,4}    {1,2,3}
                          {1,3,4}    {1,2,4}
                          {2,3,4}    {1,2,5}
                          {1,2,3,4}  {1,3,4}
                                     {1,3,5}
                                     {1,4,5}
                                     {2,3,4}
                                     {2,4,5}
                                     {1,2,3,4}
                                     {1,2,3,5}
                                     {1,2,4,5}
                                     {1,3,4,5}
                                     {2,3,4,5}
                                     {1,2,3,4,5}

Multisets of this type are ranked by A355529, complement A368100.
For divisors instead of factors we have A355740, complement A368110.
The complement for set-systems is A367902, ranks A367906, unlabeled A368095.
The version for set-systems is A367903, ranks A367907, unlabeled A368094.
For non-isomorphic multiset partitions we have A368097, complement A368098.
The version for factorizations is A368413, complement A368414.
The complement is counted by A370582.
For a unique choice we have A370584.
Partial sums of A370587, complement A370586.
The minimal case is A370591.
The version for partitions is A370593, complement A370592.
For binary indices instead of factors we have A370637, complement A370636.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A001055, A001414, A003963, A005117, A045778, A355739, A355745, A367867, A367905, A368187.

Table[Length[Select[Subsets[Range[n]], Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]],{n,0,10}]

a(n) = 2^n - A370582(n).

a(19)-a(34) from Alois P. Heinz, Feb 27 2024

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 39, 61, 122, 203, 315, 469, 676, 952, 1307, 1771, 3542, 5708, 8432, 11877, 16123, 21415, 27835, 35757, 45343, 57010, 70778, 87384, 106479, 129304, 155802, 187223, 374446, 588130, 835800, 1124981, 1456282, 1841361, 2281772, 2791896, 3367162

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

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.

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {1,3,4}
                         {2,3,4}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

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 are counted by A368414/A370814, complement A368413/A370813.
For prime indices we have A370582, differences A370586.
The complement for prime indices is A370583, differences A370587.
The complement is A370637, differences A370589, without ones A370643.
The case of a unique choice is A370638, maxima A370640, differences A370641.
First differences are A370639.
The minimal case of the complement is A370642, without ones A370644.
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, A367772, A367905, A367909, A367912, A368095, A368109.

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

a(2^n - 1) = A367902(n).

Partial sums of A370639.

a(19)-a(40) from Alois P. Heinz, Mar 09 2024

A367771 Number of ways to choose a different prime index of each prime index of 2n + 1.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Dec 12 2023

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.

			The prime indices of prime indices of 427 = 2*213 + 1 are {{1,1},{1,2,2}}, with four ways to choose (1,2), so a(213) = 4.
The prime indices of prime indices of 1469 = 2*734 + 1 are {{1,2},{1,2,3}}, with four choices (1,2), (1,3), (2,1), (2,3), so a(734) = 4.

The "extended" version below includes alternating zeros at even positions.
Extended positions of zeros are A355529, binary A367907.
The extended version for binary indices is A367905.
Extended positions of nonzeros are A368100, binary A367906.
Extended positions of ones are A368101, binary A367908.
The extended version without distinctness is A355741, for multisets A355744.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A007716, A355739, A355740, A355745, A367901, A367902, A367903.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[prix/@prix[2n+1]], UnsameQ@@#&]],{n,0,100}]

A370585 Number of maximal subsets of {1..n} such that it is possible to choose a different prime factor of each element.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 5, 7, 11, 25, 25, 38, 38, 84, 150, 178, 178, 235, 235, 341, 579, 1235, 1235, 1523, 1968, 4160, 4824, 6840, 6840, 9140, 9140, 10028, 16264, 33956, 48680, 56000, 56000, 116472, 186724, 223884, 223884, 290312, 290312, 403484, 484028, 1001420

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

First differs from A307984 at a(21) = 579, A307984(21) = 578. The difference is due to the set {10,11,13,14,15,17,19,21}, which is not a basis because log(10) + log(21) = log(14) + log(15).

Also length-pi(n) subsets of {1..n} such that it is possible to choose a different prime factor of each element.

			The a(0) = 1 through a(8) = 7 subsets:
  {}  {}  {2}  {2,3}  {2,3}  {2,3,5}  {2,3,5}  {2,3,5,7}  {2,3,5,7}
                      {3,4}  {3,4,5}  {2,5,6}  {2,5,6,7}  {2,5,6,7}
                                      {3,4,5}  {3,4,5,7}  {3,4,5,7}
                                      {3,5,6}  {3,5,6,7}  {3,5,6,7}
                                      {4,5,6}  {4,5,6,7}  {3,5,7,8}
                                                          {4,5,6,7}
                                                          {5,6,7,8}

Multisets of this type are ranked by A368100, complement A355529.
Factorizations of this type are counted by A368414, complement A368413.
The version for set-systems is A368601, max of A367902 (complement A367903).
This is the maximal case of A370582, complement A370583, cf. A370584.
A different kind of maximality is A370586, complement A370587.
The case containing n is A370590, complement A370591.
Partitions of this type (choosable) are A370592, complement A370593.
For binary indices instead of factors we have A370640, cf. A370636, A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A005117, A045778, A133686, A333331, A355739, A355740, A355744, A355745, A367905, A368110.

Table[Length[Select[Subsets[Range[n], {PrimePi[n]}],Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,10}]

More terms from Jinyuan Wang, Feb 14 2025

cp's OEIS Frontend

A140638 Number of connected graphs on n labeled nodes that contain at least two cycles.

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A355737 Number of ways to choose a sequence of divisors, one of each prime index of n (with multiplicity), such that the result has no common divisor > 1.

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A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

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A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.

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A370320 Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.

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A370814 Number of condensed integer factorizations of n into unordered factors > 1.

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A370583 Number of subsets of {1..n} such that it is not possible to choose a different prime factor of each element.

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A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

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