A304360 -id:A304360 - cp's OEIS frontend

A324741 Number of subsets of {1...n} containing no prime indices of the elements.

1, 2, 3, 5, 8, 13, 19, 30, 54, 96, 156, 248, 440, 688, 1120, 1864, 3664, 5856, 11232, 16896, 31296, 53952, 91008, 137472, 270528, 516720, 863088, 1710816, 3173856, 4836672, 9329472, 14897376, 29788128, 52256448, 88429248, 166037184, 331648704, 497685888, 829449600

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(6) = 19 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}    {2,4}
                              {3,4}    {2,5}
                              {4,5}    {3,4}
                              {2,4,5}  {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {2,4,5}
                                       {3,4,6}
                                       {4,5,6}
An example for n = 20 is {5,6,7,9,10,12,14,15,16,19,20}, with prime indices:
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  19: {8}
  20: {1,1,3}
None of these prime indices {1,2,3,4,8} belong to the subset, as required.

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

The maximal case is A324743. The strict integer partition version is A324751. The integer partition version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324742.

Table[Length[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]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(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

A324847 Numbers divisible by at least one of their prime indices.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

If n is in the sequence, then so are all multiples of n. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}

Robert Israel, Table of n, a(n) for n = 1..10000

Complement of A324846.
Cf. A003963, A056239, A112798, A120383, A289509, A290822, A304360, A306844.
Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324848, A324849, A324850, A324852, A324853.

filter:= proc(n) local F;
  F:= map(numtheory:-pi, numtheory:-factorset(n));
  ormap(t -> n mod t = 0, F);
end proc:
select(filter, [$1..200]); # Robert Israel, Mar 19 2019

Select[Range[100],Or@@Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>Divisible[#,PrimePi[p]]]&]

isok(n) = {my(f = factor(n)[,1]); for (k=1, #f, if (!(n % primepi(f[k])), return (1));); return (0);} \\ Michel Marcus, Mar 19 2019

A324525 Numbers divisible by prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 27, 32, 36, 54, 64, 72, 81, 108, 125, 128, 144, 162, 216, 243, 250, 256, 288, 324, 432, 486, 500, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1125, 1152, 1250, 1296, 1458, 1728, 1944, 2000, 2048, 2187, 2250, 2304, 2401, 2500, 2592

Offset: 1

Gus Wiseman, Mar 08 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 where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   54: {1,2,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}

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

Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324524, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=PrimePi[p]]&]
seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020

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

Original entry on oeis.org

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]

A324765 Number of recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 26, 52, 119, 266, 618, 1432, 3402, 8093, 19505, 47228, 115244, 282529, 696388, 1723400

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of a terminal subtree is a branch of the same subtree.

			The a(1) = 1 through a(6) = 11 recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          (o((o)))   (o((oo)))
                          ((((o))))  (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Cf. A000081, A290689, A301700, A304360, A306844, A317787, A318185.
Cf. A324695, A324751, A324756, A324758, A324764, A324766, A324767, A324768.
Cf. A324838, A324840, A324844.

nallt[n_]:=Select[Union[Sort/@Join@@(Tuples[nallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@#,#]=={}&];
Table[Length[nallt[n]],{n,10}]

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

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[#]]&]

A324840 Number of fully recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 14, 23, 46, 85, 165, 313, 625, 1225, 2459, 4919, 9928, 20078, 40926, 83592

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

			The a(1) = 1 through a(7) = 14 fully recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((((oo))))   ((oo)(oo))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o))(oo))
                                                  (((o)(oo)))
                                                  ((o)((oo)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))

Gus Wiseman, The a(8) = 23 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(9) = 46 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(10) = 85 fully recursively anti-transitive rooted trees.

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324768, A324769, A324770.
Cf. A324838, A324841, A324844, A324846.

dallt[n_]:=Select[Union[Sort/@Join@@(Tuples[dallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&];
Table[Length[dallt[n]],{n,10}]

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

cp's OEIS Frontend

A324741 Number of subsets of {1...n} containing no prime indices of the elements.

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A324847 Numbers divisible by at least one of their prime indices.

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A324525 Numbers divisible by prime(k)^k for each prime index k.

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

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A324765 Number of recursively anti-transitive rooted trees with n nodes.

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

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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.