A276625 -id:A276625 - cp's OEIS frontend

A324751 Number of strict integer partitions of n containing no prime indices of the parts.

1, 1, 1, 1, 2, 1, 3, 3, 2, 4, 5, 5, 6, 8, 8, 12, 10, 14, 13, 18, 19, 26, 25, 30, 34, 39, 40, 51, 55, 60, 71, 77, 90, 97, 111, 123, 136, 153, 170, 179, 216, 230, 264, 282, 322, 345, 385, 423, 470, 513, 573, 629, 686, 755, 834, 910, 1005, 1095, 1194, 1303, 1433

Offset: 0

Gus Wiseman, Mar 16 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(13) = 8 strict integer partitions (A...D = 10...13):
  1   2   3   4    5   6    7    8    9    A    B     C     D
              31       42   43   71   54   64   65    75    76
                       51   52        63   73   83    84    85
                                      72   82   542   93    94
                                           91   731   A2    B2
                                                      B1    643
                                                            751
                                                            931

The subset version is A324741, with maximal case A324743. The non-strict version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324742, A324748.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421

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.

Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,4}      {1,4}        {1,4}
                           {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,6}
                                      {1,2,3,5}    {1,2,3,4}
                                      {1,2,3,4,5}  {1,2,3,5}
                                                   {1,2,3,6}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.

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

The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
Cf. A324697, A324737, A324741, A324743.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{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(!bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 15 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 15 2019

A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 43, 89, 185, 391, 840, 1822, 3975, 8727, 19280, 42841, 95661, 214490

Offset: 1

Gus Wiseman, Jul 04 2018

			The a(7) = 11 locally disjoint rooted identity trees:
((((((o))))))
((((o(o)))))
(((o((o)))))
((o(((o)))))
((o(o(o))))
(((o)((o))))
(o((((o)))))
(o((o(o))))
(o(o((o))))
((o)(((o))))
(o(o)((o)))

Gus Wiseman, The a(8) = 21 locally disjoint rooted identity trees.

Cf. A000081, A004111, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316473, A316474, A316494.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&(Intersection@@#=!={})&]=={}&]];
Table[Length[strut[n]],{n,20}]

A324572 Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 2, 0, 4, 1, 2, 1, 4, 1, 3, 1, 5, 3, 5, 1, 6, 2, 6, 1, 7, 2, 7, 2, 11, 4, 8, 3, 11, 5, 10, 4, 13, 5, 11, 5, 16, 8, 14, 5, 19, 8, 18, 6, 22, 8, 22, 7, 26, 10, 25, 8, 33, 12, 29, 11, 36, 13, 34, 12, 40, 16, 41, 14, 47, 17, 45, 16, 55

Offset: 0

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing partitions (cf. A001462, A304679).
The Heinz numbers of these partitions are given by A324571.
The case where the distinct parts are taken in increasing order is counted by A033461, with Heinz numbers given by A109298.

			The first 19 terms count the following integer partitions:
   1: (1)
   4: (22)
   4: (211)
   6: (3111)
   8: (41111)
   9: (333)
  10: (511111)
  10: (322111)
  12: (6111111)
  12: (4221111)
  12: (33222)
  14: (71111111)
  14: (52211111)
  16: (811111111)
  16: (622111111)
  16: (4444)
  16: (442222)
  17: (43331111)
  18: (9111111111)
  18: (7221111111)
  19: (533311111)

Cf. A001156, A033461, A109298, A117144, A276078, A324524, A324571.

Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Table[Length[Select[IntegerPartitions[n],Union[#]==Length/@Split[#]&]],{n,0,30}]

More terms from Alois P. Heinz, Mar 08 2019

A324704 Lexicographically earliest sequence containing 1 and all numbers > 2 divisible by prime(m) for some m already in the sequence.

Original entry on oeis.org

1, 4, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84

Offset: 1

Gus Wiseman, Mar 11 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:
   1: {}
   4: {1,1}
   6: {1,2}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}

Complement of A324698.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

aQ[n_]:=Switch[n,1,True,2,False,,!And@@Cases[FactorInteger[n],{p,k_}:>!aQ[PrimePi[p]]]];
Select[Range[100],aQ]

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882

Offset: 1

Gus Wiseman, Mar 17 2019

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (o = {}):
  {{{{{{{o}}}}}}}
  {{{{{o,{o}}}}}}
  {{{{o,{{o}}}}}}
  {{{o,{{{o}}}}}}
  {{{o,{o,{o}}}}}
  {{{{o},{{o}}}}}
  {{o,{{{{o}}}}}}
  {{o,{{o,{o}}}}}
  {{o,{o,{{o}}}}}
  {{{o},{{{o}}}}}
  {{{o},{o,{o}}}}
  {{o,{o},{{o}}}}
  {o,{{{{{o}}}}}}
  {o,{{{o,{o}}}}}
  {o,{{o,{{o}}}}}
  {o,{{o},{{o}}}}
  {{o},{{{{o}}}}}
  {{o},{{o,{o}}}}
  {{o},{o,{{o}}}}
  {{{o}},{o,{o}}}

			The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
  o  (o)  ((o))  (((o)))  ((o(o)))   (((o(o))))   ((o(o(o))))
                          (o((o)))   ((o((o))))   (o((o(o))))
                          ((((o))))  (o(((o))))   ((((o(o)))))
                                     (((((o)))))  (((o)((o))))
                                                  (((o((o)))))
                                                  ((o)(((o))))
                                                  ((o(((o)))))
                                                  (o((((o)))))
                                                  ((((((o))))))

Gus Wiseman, The a(9) = 41 anti-transitive rooted identity trees.

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.

Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

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

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A063895 Start with x, xy; then concatenate each word in turn with all preceding words, getting x xy xxy xxxy xyxxy xxxxy xyxxxy xxyxxxy ...; sequence gives number of words of length n. Also binary trees by degree: x (x,y) (x,(x,y)) (x,(x,(x,y))) ((x,y),(x,(x,y)))...

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 11, 22, 43, 88, 179, 372, 774, 1631, 3448, 7347, 15713, 33791, 72923, 158021, 343495, 749102, 1638103, 3591724, 7893802, 17387931, 38379200, 84875596, 188036830, 417284181, 927469845, 2064465341, 4601670625, 10270463565, 22950838755

Offset: 1

Claude Lenormand (claude.lenormand(AT)free.fr), Aug 29 2001

Also binary rooted identity trees (those with no symmetries, cf. A004111).
From Gus Wiseman, May 04 2021: (Start)
Also the number of unlabeled binary rooted semi-identity trees with 2*n - 1 nodes. In a semi-identity tree, only the non-leaf branches directly under any given vertex are required to be distinct. Alternatively, an unlabeled rooted tree is a semi-identity tree iff the non-leaf branches of the root are all distinct and are themselves semi-identity trees. For example, the a(3) = 1 through a(6) = 6 trees are:
  (o(oo))  (o(o(oo)))  ((oo)(o(oo)))  ((oo)(o(o(oo))))  ((o(oo))(o(o(oo))))
                       (o(o(o(oo))))  (o((oo)(o(oo))))  ((oo)((oo)(o(oo))))
                                      (o(o(o(o(oo)))))  ((oo)(o(o(o(oo)))))
                                                        (o((oo)(o(o(oo)))))
                                                        (o(o((oo)(o(oo)))))
                                                        (o(o(o(o(o(oo))))))
The a(8) = 11 trees with 15 nodes:
  ((o(oo))((oo)(o(oo))))
  ((o(oo))(o(o(o(oo)))))
  ((oo)((oo)(o(o(oo)))))
  ((oo)(o((oo)(o(oo)))))
  ((oo)(o(o(o(o(oo))))))
  (o((o(oo))(o(o(oo)))))
  (o((oo)((oo)(o(oo)))))
  (o((oo)(o(o(o(oo))))))
  (o(o((oo)(o(o(oo))))))
  (o(o(o((oo)(o(oo))))))
  (o(o(o(o(o(o(oo)))))))
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to rooted trees

Cf. A063894, A036774.
The non-semi-identity version is 2*A001190(n)-1, ranked by A111299.
Semi-binary trees are also counted by A001190, but ranked by A292050.
The not necessarily binary version is A306200, ranked A306202.
The Matula-Goebel numbers of these trees are A339193.
The plane tree version is A343663.
A000081 counts unlabeled rooted trees with n nodes.
A004111 counts identity trees, ranked by A276625.
A306201 counts balanced semi-identity trees, ranked by A306203.
A331966 counts lone-child avoiding semi-identity trees, ranked by A331965.
Cf. A001678, A331934, A331963, A331964.

a:= proc(n) option remember; `if`(n<3, n*(3-n)/2, add(a(i)*a(n-i),
      i=1..(n-1)/2)+`if`(irem(n, 2, 'r')=0, (p->(p-1)*p/2)(a(r)), 0))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 02 2013

a[n_] := a[n] = If[n<3, n*(3-n)/2, Sum[a[i]*a[n-i], {i, 1, (n-1)/2}]+If[{q, r} = QuotientRemainder[n, 2]; r == 0, (a[q]-1)*a[q]/2, 0]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)
ursiq[n_]:=Join@@Table[Select[Union[Sort/@Tuples[ursiq/@ptn]],#=={}||#=={{},{}}||Length[#]==2&&(UnsameQ@@DeleteCases[#,{}])&],{ptn,IntegerPartitions[n-1]}];Table[Length[ursiq[n]],{n,1,15,2}] (* Gus Wiseman, May 04 2021 *)

{a(n)=local(A, m); if(n<1, 0, m=1; A=O(x); while( m<=n, m*=2; A=1-sqrt(1-2*x-2*x^2+subst(A, x, x^2))); polcoeff(A, n))}

a(n) = (sum a(i)*a(j), i+j=n, i2. a(1)=a(2)=1.
G.f. A(x) = 1-sqrt(1-2x-2x^2+A(x^2)) satisfies x+x^2-A(x)+(A(x)^2-A(x^2))/2=0, A(0)=0. - Michael Somos, Sep 06 2003
a(n) ~ c * d^n / n^(3/2), where d = 2.33141659246516873904600076533362924695..., c = 0.2873051160895040470174351963... . - Vaclav Kotesovec, Sep 11 2014

Additional comments and g.f. from Christian G. Bower, Nov 29 2001

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 152, 172, 212, 214, 224, 256, 262, 304, 326, 344, 424, 428, 448, 512, 524, 526, 608, 622, 652, 688, 766, 848, 856, 886, 896, 1024, 1048, 1052, 1154, 1216, 1226, 1244, 1304, 1376, 1438, 1532, 1696

Offset: 1

Gus Wiseman, Jan 30 2020

nonn

Also Matula-Goebel numbers of lone-child-avoiding rooted trees at with at most one non-leaf branch under any given vertex. A rooted tree is lone-child-avoiding if there are no unary branchings. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Also Matula-Goebel numbers of lone-child-avoiding locally disjoint semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct.

			The sequence of all lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
  224: (ooooo(oo))

Robert Israel, Table of n, a(n) for n = 1..3140

These trees counted by number of vertices are A212804.
The semi-lone-child-avoiding version is A331681.
The non-semi-identity version is A331871.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Unlabeled semi-identity trees are counted by A306200, with Matula-Goebel numbers A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
Cf. A000081, A007097, A061775, A196050, A276625, A316470, A316696, A331678, A331679, A331680, A331682, A331686, A331687.

N:= 10^4: # for terms <= N
S:= {1}:
with(queue):
Q:= new(1):
while not empty(Q) do
  r:= dequeue(Q);
  p:= ithprime(r);
  newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;
  S:= S union newS;
  for s in newS do enqueue(Q,s) od:
od:
sort(convert(S,list)); # Robert Israel, Feb 05 2020

uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,},{p,1}}/;uryQ[PrimePi[p]])|({{2,k_}}/;k>1)];
Select[Range[100],uryQ]

Intersection of A291636, A316495, and A306202.

A324698 Lexicographically earliest sequence containing 2 and all numbers > 1 whose prime indices already belong to the sequence.

Original entry on oeis.org

2, 3, 5, 9, 11, 15, 23, 25, 27, 31, 33, 45, 47, 55, 69, 75, 81, 83, 93, 97, 99, 103, 115, 121, 125, 127, 135, 137, 141, 155, 165, 197, 207, 211, 225, 235, 243, 249, 253, 257, 275, 279, 291, 297, 309, 341, 345, 347, 363, 375, 379, 381, 405, 411, 415, 419, 423

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}
   9: {2,2}
  11: {5}
  15: {2,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  45: {2,2,3}
  47: {15}
  55: {3,5}
  69: {2,9}
  75: {2,3,3}
  81: {2,2,2,2}
  83: {23}
  93: {2,11}
  97: {25}
  99: {2,2,5}

Complement of A324704.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

aQ[n_]:=Switch[n,1,False,2,True,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[100],aQ]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A324751 Number of strict integer partitions of n containing no prime indices of the parts.

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A324736 Number of subsets of {1...n} containing all prime indices of the elements.

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A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

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A324572 Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.

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A324704 Lexicographically earliest sequence containing 1 and all numbers > 2 divisible by prime(m) for some m already in the sequence.

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A324764 Number of anti-transitive rooted identity trees with n nodes.

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A063895 Start with x, xy; then concatenate each word in turn with all preceding words, getting x xy xxy xxxy xyxxy xxxxy xyxxxy xxyxxxy ...; sequence gives number of words of length n. Also binary trees by degree: x (x,y) (x,(x,y)) (x,(x,(x,y))) ((x,y),(x,(x,y)))...

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A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

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