A355744 -id:A355744 - cp's OEIS frontend

A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 4, 1, 2, 3, 4, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 1, 4, 2, 6, 3, 6, 4, 7, 2, 2, 5, 4, 2, 6, 3, 4, 2, 6, 3, 4, 4, 5, 4, 4, 3, 7, 4, 2, 4, 6, 2, 7, 1, 7, 4, 2, 2, 6, 6, 6, 3, 4, 6, 6, 4, 6, 7, 4, 2, 5, 2, 2, 5, 4, 4, 7

Offset: 1

Gus Wiseman, Jul 16 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(15) = 4 multisets are: {1,1}, {1,2}, {1,3}, {2,3}.
The a(18) = 3 multisets are: {1,1,1}, {1,1,2}, {1,2,2}.

Robert Price, Table of n, a(n) for n = 1..2000
Wikipedia, Axiom of choice.

Counting all choices of divisors gives A355731, firsts A355732.
Positions of first appearances are A355734.
Choosing weakly increasing divisors gives A355735, firsts A355736.
Choosing only prime divisors gives A355744.
The version choosing a divisor of each number from 1 to n is A355747.
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.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A340852 lists numbers that can be factored into divisors of bigomega.
Cf. A000720, A076610, A289509, A316524, A335433, A335448, A340606, A344616, A355737, A355739, A355740, A355741, A355746.

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

A368109 Number of ways to choose a binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 4, 4, 4, 4, 8, 8, 8, 8, 3, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12

Offset: 0

Gus Wiseman, Dec 12 2023

nonn

First differs from A367912 at a(52) = 8, A367912(52) = 7.
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}.
Run-lengths are all 4 or 8.

			The binary indices of binary indices of 20 are {{1,2},{1,3}}, with choices (1,1), (1,3), (2,1), (2,3), so a(20) = 4.
The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,1,1), (1,1,3), (1,3,2), (1,3,3), (2,1,2), (2,1,3), (2,3,2), (2,3,3), so a(52) = 8.

All entries appear to belong to A003586.
Positions of ones are A253317.
The version for prime indices is A355741, for multisets A355744.
Choosing a multiset (not sequence) gives A367912, firsts A367913.
Positions of first appearances are A368111, sorted A368112.
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, A326753, A355731, A355739, A367771, A367905, A367906, A367915.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Table[Length[Tuples[bpe/@bpe[n]]], {n,0,100}]

a(n) = Product_{k in A048793(n)} A000120(k).

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}]

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

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}]

A355742 Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 20 2022

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 49 are {4,4}, and the a(49) = 4 choices are: (2,2), (2,4), (4,2), (4,4).
The prime indices of 777 are {2,4,12}, and the a(777) = 6 choices are: (2,2,2), (2,2,3), (2,2,4), (2,4,2), (2,4,3), (2,4,4).

Wikipedia, Cartesian product.

The unordered version is A001970, row-sums of A061260.
Positions of 1's are A076610, just primes A355743.
Positions of 0's are A299174.
Allowing all divisors (not just primes) gives A355731, firsts A355732.
Choosing only prime factors (not prime-powers) gives A355741.
Counting multisets of primes gives A355744.
The case of weakly increasing primes A355745, all divisors A355735.
A000688 counts factorizations into prime powers.
A001414 adds up distinct prime factors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A120383, A279784, A289509, A324850, A355733, A355739, A355746.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@PrimeOmega/@primeMS[n],{n,100}]

Totally multiplicative with a(prime(k)) = A001222(k).

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

A370582 Number of 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, 2, 4, 6, 12, 20, 40, 52, 72, 116, 232, 320, 640, 1020, 1528, 1792, 3584, 4552, 9104, 12240, 17840, 27896, 55792, 67584, 83968, 130656, 150240, 198528, 397056, 507984, 1015968, 1115616, 1579168, 2438544, 3259680, 3730368, 7460736, 11494656, 16145952, 19078464, 38156928

Offset: 0

Gus Wiseman, Feb 25 2024

nonn

			The a(0) = 1 through a(6) = 20 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {6}
                             {2,5}    {2,3}
                             {3,4}    {2,5}
                             {3,5}    {2,6}
                             {4,5}    {3,4}
                             {2,3,5}  {3,5}
                             {3,4,5}  {3,6}
                                      {4,5}
                                      {4,6}
                                      {5,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

The version for set-systems is A367902, ranks A367906, unlabeled A368095.
The complement for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368098, complement A368097.
Multisets of this type are ranked by A368100, complement A355529.
For divisors instead of factors we have A368110, complement A355740.
The version for factorizations is A368414, complement A368413.
The complement is counted by A370583.
For a unique choice we have A370584.
The maximal case is A370585.
Partial sums of A370586, complement A370587.
The version for partitions is A370592, complement A370593.
For binary indices instead of factors we have A370636, complement 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, A001055, A001414, A003963, A005117, A045778, A133686, A355739, A355744, A355745, A367905.

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

a(p) = 2 * a(p-1) for prime p. - David A. Corneth, Feb 25 2024

a(n) = 2^n - A370583(n).

a(19) from David A. Corneth, Feb 25 2024

a(20)-a(41) from Alois P. Heinz, Feb 25 2024

A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 4, 4, 4, 4, 7, 7, 7, 7, 3, 3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 8, 8, 8, 8

Offset: 0

Gus Wiseman, Dec 12 2023

nonn

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}.

The run-lengths are all 4 or 8.

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with multiset choices {1,1,2}, {1,1,3}, {1,2,2}, {1,2,3}, {1,3,3}, {2,2,3}, {2,3,3}, so a(52) = 7.

Positions of ones are A253317.
The version for multisets and divisors is A355733, for sequences A355731.
The version for multisets is A355744, for sequences A355741.
For a sequence of distinct choices we have A367905, firsts A367910.
Positions of first appearances are A367913, sorted A367915.
Choosing a sequence instead of multiset gives A368109, firsts A368111.
Choosing a set instead of multiset gives A368183, firsts A368184.
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, A326753, A355735, A355739, A355740, A355745, A367771, A367906.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]], {n,0,100}]

A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 18, 36, 48, 68, 104, 208, 284, 568, 888, 1296, 1548, 3096, 3968, 7936, 10736, 15440, 24008, 48016, 58848, 73680, 114368, 132608, 176240, 352480, 449824, 899648, 994976, 1399968, 2160720, 2859584, 3296048, 6592096, 10156672, 14214576, 16892352

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3).

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {2,3}
                             {2,5}    {2,5}
                             {3,4}    {2,6}
                             {3,5}    {3,4}
                             {4,5}    {3,5}
                             {2,3,5}  {3,6}
                             {3,4,5}  {4,5}
                                      {4,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

For divisors instead of factors we have A051026, cf. A368110, A355740.
The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
For existence we have A370582, differences A370586.
For nonexistence we have A370583, differences A370587.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, cf. A370592, A370593.
For binary indices instead of factors we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
For unlabeled multiset partitions we have A370646, cf. A368098, A368097.
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, A005117, A045778, A133686, A307984, A355739, A355744, A355745, A367905.

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

More terms from Jinyuan Wang, Mar 28 2025

cp's OEIS Frontend

A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.

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A368109 Number of ways to choose a binary index of each binary index of n.

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

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A355742 Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

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

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A370582 Number of subsets of {1..n} such that it is possible to choose a different prime factor of each element.

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