A001462 -id:A001462 - cp's OEIS frontend

A324697 Lexicographically earliest sequence of positive integers > 1 that are prime or whose prime indices already belong to the sequence.

2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 41, 43, 45, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 75, 79, 81, 83, 85, 89, 93, 97, 99, 101, 103, 107, 109, 113, 115, 121, 123, 125, 127, 131, 135, 137, 139, 141, 149, 151, 153, 155, 157, 163, 165

Offset: 1

Gus Wiseman, Mar 10 2019

nonn

A self-describing sequence, similar to A304360.

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 sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  37: {12}
  41: {13}
  43: {14}
  45: {2,2,3}

Complement of A324705.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324698, A324699, A324700, A324701, A324702, A324703, A324704.

aQ[n_]:=Switch[n,1,False,?PrimeQ,True,,And@@Cases[FactorInteger[n],{p_,k_}:>aQ[PrimePi[p]]]];
Select[Range[100],aQ]

A054354 First differences of Kolakoski sequence A000002.

Original entry on oeis.org

1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1, 0, 1, 0, -1, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1

Offset: 1

N. J. A. Sloane, May 07 2000

sign

The Kolakoski sequence has only 1's and 2's, and is cubefree. Thus, for all n>=1, a(n) is in {-1, 0, 1}, a(n+1) != a(n), and if a(n) = 0, a(n+1) = -a(n-1), while if a(n) != 0, either a(n+1) = 0 and a(n+2) = -a(n) or a(n+1) = -a(n). A further consequence is that the maximum gap between equal values is 4: for all n, there is an integer k, 1Jean-Christophe Hervé, Oct 05 2014
From Daniel Forgues, Jul 07 2015: (Start)
Second differences: {-1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, ...}
The sequence of first differences bounces between -1 and 1 with a slope whose absolute value is either 1 or 2. We can compress the information in the second differences into {-1, 1, -2, 2, -1, 2, -1, 1, ...} since the -1 and the 1 come in pairs; which can be compressed further into {1, 1, 2, 2, 1, 2, 1, 1, ...} since the signs alternate, where we only need to know that the initial sign is negative. (End)
This appears to divide the positive integers into three sets, each with density approaching 1/3. Note there are no adjacent equal parts (as mentioned above). - Gus Wiseman, Oct 10 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000002, A054353.
Positions of 0 are A078649.
For Golomb's sequence (A001462) we have A088517.
Positions of -1 are A156242 (descents).
Positions of 1 are A156243 (ascents).
First differences (or second differences of A000002) are A376604.
The Kolakoski sequence (A000002):
- Restrictions: A074264, A100428, A100429, A156263, A156264, A288605.
- Patterns: A013947, A013948, A022292, A074262, A074263.
- Statistics: A074286, A088568, A156077, A156253.
- Transformations: A054354, A156728, A306323, A332273, A332875, A333229.
Cf. A333254.

a054354 n = a054354_list !! (n-1)
a054354_list = zipWith (-) (tail a000002_list) a000002_list
-- Reinhard Zumkeller, Aug 03 2013

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 70}, {a2[[n]]}]; Differences[a2] (* Jean-François Alcover, Jun 18 2013 *)

Abs(a(n)) = (A000002(n)+A000002(n+1)) mod 2. - Benoit Cloitre, Nov 17 2003

A324743 Number of maximal subsets of {1...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 8, 8, 8, 8, 12, 12, 18, 18, 19, 19, 30, 30, 54, 54, 54, 54, 96, 96, 96, 96, 96, 96, 156, 156, 244, 244, 248, 248, 248, 248, 440, 440, 440, 440, 688, 688, 1120, 1120, 1120, 1120, 1864, 1864, 1864, 1864, 1864, 1864, 3664, 3664, 3664, 3664, 3664

Offset: 0

Gus Wiseman, Mar 15 2019

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(0) = 1 through a(8) = 8 maximal subsets:
  {}  {1}  {1}  {2}    {1,3}  {1,3}    {1,3}    {1,3,7}  {1,3,7}
           {2}  {1,3}  {2,4}  {1,5}    {1,5}    {1,5,7}  {1,5,7}
                       {3,4}  {3,4}    {2,4,5}  {2,4,5}  {2,4,5,8}
                              {2,4,5}  {3,4,6}  {2,5,7}  {2,5,7,8}
                                       {4,5,6}  {3,4,6}  {3,4,6,8}
                                                {3,6,7}  {3,6,7,8}
                                                {4,5,6}  {4,5,6,8}
                                                {5,6,7}  {5,6,7,8}
An example for n = 15 is {1,5,7,9,13,15}, with prime indices:
  1: {}
  5: {3}
  7: {4}
  9: {2,2}
  13: {6}
  15: {2,3}
None of these prime indices {2,3,4,6} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The non-maximal case is A324741. The case for subsets of {2...n} is A324763.
Cf. A000720, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844, A320426, A324764.
Cf. A324695, A324736, A324742, A324744, A324751, A324756, A324758.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{n,0,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b,k), e=bitor(e,p[k]), if(!bittest(e,k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

A324753 Number of integer partitions of n containing all prime indices of their parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 8, 14, 16, 23, 29, 40, 49, 66, 81, 109, 133, 172, 211, 274, 332, 419, 511, 640, 775, 965, 1165, 1434, 1730, 2109, 2530, 3083, 3683, 4447, 5308, 6375, 7573, 9062, 10730, 12786, 15104, 17909, 21095, 24937, 29284, 34488, 40421, 47450

Offset: 0

Gus Wiseman, Mar 16 2019

nonn

These could be described as transitive integer partitions.

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(1) = 1 through a(8) = 8 integer partitions:
  (1)  (11)  (21)   (211)   (41)     (321)     (421)      (3221)
             (111)  (1111)  (221)    (411)     (2221)     (4211)
                            (2111)   (2211)    (3211)     (22211)
                            (11111)  (21111)   (4111)     (32111)
                                     (111111)  (22111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

The subset version is A324736. The strict case is A324748. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A000837, A001462, A007097, A051424, A112798, A276625, A279861, A290689, A290760.
Cf. A324697, A324737.

Table[Length[Select[IntegerPartitions[n],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

A156253 Least k such that A054353(k) >= n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 47, 48, 49, 50

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

a(n)=1 plus the number of symbol changes in the first n terms of A000002. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010
From N. J. A. Sloane, Nov 12 2018: (Start)
This seems to be A001462 rewritten so the run lengths are given by A000002. The companion sequence, A000002 rewritten so the run lengths are given by A001462, is A321020.
Note that Kolakoski's sequence A000002 and Golomb's sequence A001462 have very similar definitions, although the asymptotic behavior of A001462 is well-understood, while that of A000002 is a mystery. The asymptotic behavior of the two hybrids A156253 and A321020 might be worth investigating. (End)
To expand upon N. J. A. Sloane's comments, it's worth noting that Golomb's sequence has a formula from Colin Mallows: g(n) = g(n-g(g(n-1))) + 1, which closely resembles a(n) = a(n-gcd(a(a(n-1)),2)) + 1. - Jon Maiga, May 16 2023

Jon Maiga, Table of n, a(n) for n = 1..10000
J. M. Fedou and G. Fici, Some remarks on differentiable sequences and recursivity, Journal of Integer Sequences 13(3): Article 10.3.2 (2010).
Jon Maiga, A Recurrence Related to the Kolakoski Sequence
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides (Mentions this sequence)

Cf. A000002, A001462, A054353, A156351, A321020.

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 80}, {i, 1, a2[[n]]}]; a3 = Accumulate[a2]; a[1] = 1; a[n_] := a[n] = For[k = a[n - 1], True, k++, If[a3[[k]] >= n, Return[k]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 18 2013 *)
a[1] = 1;
a[n_]:=a[n]=a[n-GCD[a[a[n - 1]], 2]]+1
Array[a, 100] (* Jon Maiga, May 16 2023 *)

Conjecture: a(n) should be asymptotic to 2n/3.
Length of n-th run of the sequence = A000002(n). - Benoit Cloitre, Feb 19 2009
Conjecture: a(n) = (a(a(n-1)) mod 2) + a(n-2) + 1. - Jon Maiga, Dec 09 2021
a(n) = a(n-gcd(a(a(n-1)), 2)) + 1. - Jon Maiga, May 16 2023

A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 162, 250, 256, 288, 324, 500, 512, 576, 648, 729, 1000, 1024, 1125, 1152, 1296, 1458, 2000, 2048, 2250, 2304, 2401, 2592, 2916, 4000, 4096, 4500, 4608, 4802, 5184, 5832, 6561, 8000, 8192, 9000, 9216

Offset: 1

Gus Wiseman, Mar 07 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
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 signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions in which every part divides its multiplicity (counted by A001156). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A062457.

			The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2).
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  162: {1,2,2,2,2}
  250: {1,3,3,3}
  256: {1,1,1,1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001156, A033461, A056239, A062457, A066328, A072873, A112798, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324525, A324570, A324571, A324572, A324587, A324588.
Range of values of A090884.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]):
select(q, [$1..10000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[k,PrimePi[p]]]&]
v = Join[{1}, Prime[(r = Range[10])]^r]; n = Length[v]; vmax = 10^4; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; Length[s] (* Amiram Eldar, Sep 30 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - Amiram Eldar, Sep 30 2020

A324571 Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.

Original entry on oeis.org

1, 2, 9, 12, 40, 112, 125, 352, 360, 675, 832, 1008, 2176, 2401, 3168, 3969, 4864, 7488, 11776, 14000, 19584, 29403, 29696, 43776, 44000, 63488, 75600, 104000, 105984, 123201, 151552, 161051, 214375, 237600, 267264, 272000, 335872, 496125, 561600, 571392, 608000

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679). The increasing case is A109298.
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 ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base.
Also Heinz numbers of the integer partitions counted by A324572. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Each finite set of positive integers determines a unique term with those prime indices. For example, corresponding to {1,2,4,5} is 1397088 = prime(1)^5 * prime(2)^4 * prime(4)^2 * prime(5)^1.

			The sequence of terms together with their prime indices begins as follows. For example, we have 40: {1,1,1,3} because 40 = prime(1) * prime(1) * prime(1) * prime(3).
      1: {}
      2: {1}
      9: {2,2}
     12: {1,1,2}
     40: {1,1,1,3}
    112: {1,1,1,1,4}
    125: {3,3,3}
    352: {1,1,1,1,1,5}
    360: {1,1,1,2,2,3}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
   1008: {1,1,1,1,2,2,4}
   2176: {1,1,1,1,1,1,1,7}
   2401: {4,4,4,4}
   3168: {1,1,1,1,1,2,2,5}
   3969: {2,2,2,2,4,4}
   4864: {1,1,1,1,1,1,1,1,8}
   7488: {1,1,1,1,1,1,2,2,6}
  11776: {1,1,1,1,1,1,1,1,1,9}
  14000: {1,1,1,1,3,3,3,4}
  19584: {1,1,1,1,1,1,1,2,2,7}

Cf. A001156, A033461, A056239, A062457, A109298, A112798, A117144, A118914, A124010, A181819, A276078.
Cf. A324524, A324525, A324572.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360, A304679.

Select[Range[1000],Reverse[PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==Last/@If[#==1,{},FactorInteger[#]]&]

A324744 Number of maximal subsets of {1...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 6, 8, 8, 11, 11, 22, 22, 22, 22, 28, 28, 44, 44, 52, 52, 76, 76, 88, 88, 96, 96, 184, 184, 240, 240, 264, 264, 296, 296, 592, 592, 592, 592, 728, 728, 1456, 1456, 1456, 1456, 2912, 2912, 3168, 3168, 3168, 3168, 5568, 5568, 5568, 5568

Offset: 0

Gus Wiseman, Mar 15 2019

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

Andrew Howroyd, Table of n, a(n) for n = 0..100

The non-maximal case is A324738. The case for subsets of {2...n} is A324762.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290822, A304360, A306844, A320426, A324764.
Cf. A324694, A324736, A324743, A324749, A324754, A324759.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]]],{n,0,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=for(k=1, #p, if(!bittest(b,k) && bitnegimply(p[k], b), my(e=bitor(b, 1<#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 27 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 27 2019

A012257 Irregular triangle read by rows: row 0 is {2}; if row n is {r_1, ..., r_k} then row n+1 is {r_k 1's, r_{k-1} 2's, r_{k-2} 3's, etc.}.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 13, 14

Offset: 0

Lionel Levine (levine(AT)ultranet.com)

I have sometimes referred to this as Lionel Levine's triangle in lectures. - N. J. A. Sloane, Mar 21 2021

The shape of each row tends to a limit curve when scaled to a fixed size. It is the same limit curve as this continuous version: start with f_0=x over [0,1]; then repeatedly reverse (1-x), integrate from zero (x-x^2/2), scale to 1 (2x-x^2) and invert (1-sqrt(1-x)). For the limit curve we have f'(0) = F(1) = lim A011784(n+2)/(A011784(n+1)*A011784(n)) ~ 0.27887706 (obtained numerically). - Martin Fuller, Aug 07 2006

			Initial rows are:
{2},
{1,1},
{1,2},
{1,1,2},
{1,1,2,3},
{1,1,1,2,2,3,4},
{1,1,1,1,2,2,2,3,3,4,4,5,6,7},
...

Reinhard Zumkeller, Rows n = 0..9 of triangle, flattened
N. J. A. Sloane and Brady Haran, The Levine Sequence, Numberphile video (2021)

Cf. A001462, A011784 (row sums), A012257, A014643, A112798, A181819, A182850-A182858, A296150, A304455.

a012257 n k = a012257_tabf !! (n-1) !! (k-1)
a012257_row n = a012257_tabf !! (n-1)
a012257_tabf = iterate (\row -> concat $
                        zipWith replicate (reverse row) [1..]) [1, 1]
-- Reinhard Zumkeller, Aug 11 2014, May 30 2012

T:= proc(n) option remember; `if`(n=0, 2, (h->
      seq(i$h[-i], i=1..nops(h)))([T(n-1)]))
    end:
seq(T(n), n=0..8);  # Alois P. Heinz, Mar 31 2021

row[1] = {1, 1}; row[n_] := row[n] = MapIndexed[ Function[ Table[#2 // First, {#1}]], row[n-1] // Reverse] // Flatten; Array[row, 7] // Flatten (* Jean-François Alcover, Feb 10 2015 *)
NestList[Flatten@ MapIndexed[ConstantArray[First@ #2, #1] &, Reverse@ #] &, {1, 1}, 6] // Flatten (* Michael De Vlieger, Jul 12 2017 *)

Sum of row n = A011784(n+2); e.g. row 5 is {1, 1, 1, 2, 2, 3, 4} and the sum of the elements is 1+1+1+2+2+3+4 = 14 = A011784(7). - Benoit Cloitre, Aug 06 2003

T(n,A011784(n+1)) = A011784(n). - Reinhard Zumkeller, Aug 11 2014

Initial row {2} added by N. J. A. Sloane, Mar 21 2021

A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 26, 42, 72, 120, 232, 376, 752, 1128, 2256, 4512, 8256, 13632, 27264, 42048, 82944, 158976, 313344, 497664, 995328, 1700352, 3350016, 5815296, 11630592, 17491968, 34983936, 56954880, 108933120, 210788352, 418258944, 804667392, 1609334784

Offset: 0

Gus Wiseman, Mar 13 2019

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(0) = 1 through a(6) = 26 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {1,3}  {4}    {4}      {4}
                       {1,3}  {5}      {5}
                       {2,4}  {1,3}    {6}
                       {3,4}  {1,5}    {1,3}
                              {2,4}    {1,5}
                              {2,5}    {1,6}
                              {3,4}    {2,4}
                              {4,5}    {2,5}
                              {2,4,5}  {2,6}
                                       {3,4}
                                       {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {1,3,6}
                                       {1,5,6}
                                       {2,4,5}
                                       {2,4,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..100

The maximal case is A324744. The case of subsets of {2...n} is A324739. The strict integer partition version is A324749. The integer partition version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324741, A324750, A324755, A324760.

Table[Length[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

cp's OEIS Frontend

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A324743 Number of maximal subsets of {1...n} containing no prime indices of the elements.

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A324753 Number of integer partitions of n containing all prime indices of their parts.

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A156253 Least k such that A054353(k) >= n.

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A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

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A324571 Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.

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A324744 Number of maximal subsets of {1...n} containing no element whose prime indices all belong to the subset.

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