A007097 -id:A007097 - cp's OEIS frontend

A292050 Matula-Goebel numbers of semi-binary rooted trees.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 39, 41, 43, 46, 47, 49, 51, 55, 58, 59, 62, 65, 69, 73, 77, 79, 82, 83, 85, 86, 87, 91, 93, 94, 97, 101, 109, 115, 118, 119, 121, 123, 127, 129, 137, 139, 141, 143, 145

Offset: 1

Gus Wiseman, Sep 08 2017

nonn

An unlabeled rooted tree is semi-binary if all out-degrees are <= 2. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

Cf. A000081, A001190, A007097, A036656, A061773, A111299, A214577, A291636.

nn=200;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
semibinQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]<=2,And@@semibinQ/@m]]];
Select[Range[nn],semibinQ]

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]

A245704 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A000040(a(n)), a(A091242(n)) = A002808(a(n)), where A000040(n) = n-th prime, A002808(n) = n-th composite number, and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomial over GF(2), respectively.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 15, 7, 10, 13, 16, 21, 25, 14, 18, 19, 22, 26, 33, 38, 24, 11, 28, 30, 34, 39, 49, 23, 55, 36, 20, 42, 45, 37, 50, 56, 69, 47, 35, 77, 52, 32, 60, 17, 64, 54, 70, 78, 94, 66, 51, 29, 105, 74, 48, 41, 84, 53, 27, 88, 76, 95, 106, 73, 125, 91, 72, 44, 140, 97, 100, 68, 58, 115, 75, 40

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

All the permutations A091203, A091205, A106443, A106445, A106447, A235042 share the same property that the binary representations of irreducible GF(2) polynomials (A014580) are mapped bijectively to the primes (A000040) but while they determine the mapping of corresponding reducible polynomials (A091242) to the composite numbers (A002808) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245703.
Cf. A000040, A002808, A007097, A010051, A014580, A091225, A091226, A091230, A091242, A091245.
Similar or related permutations: A245701, A245822, A227413, A091203, A091205, A106443, A106445, A106447, A235042, A244987, A245450.

allocatemem(123456789);
default(primelimit, 2^22)
A091226 = vector(2^22);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
j=0; n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
for(n=1, 10001, write("b245704.txt", n, " ", A245704(n)));

;; With memoization-macro definec.
(definec (A245704 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (A000040 (A245704 (A091226 n)))) (else (A002808 (A245704 (A091245 n))))))

a(1) = 1, after which, if A091225(n) is 1 [i.e. n is in A014580], then a(n) = A000040(a(A091226(n))), otherwise a(n) = A002808(a(A091245(n))).
As a composition of related permutations:
a(n) = A227413(A245701(n)).
a(n) = A245822(A091205(n)).
Other identities. For all n >= 1, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A091205 has the same property].
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials (= A014580) to primes and the corresponding representations of reducible polynomials to composites].

A306202 Matula-Goebel numbers of rooted semi-identity trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 84, 85

Offset: 1

Gus Wiseman, Jan 29 2019

nonn

Definition: A positive integer belongs to the sequence iff its prime indices greater than 1 are distinct and already belong to the sequence. 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 all unlabeled rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A004111, A007097, A276625, A277098, A306200, A306201, A316467.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n],1],And@@psidQ/@primeMS[n]];
Select[Range[100],psidQ]

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}]

A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 25, 12, 19, 22, 9, 16, 47, 10, 31, 28, 29, 50, 13, 24, 21, 38, 15, 44, 61, 18, 137, 32, 43, 94, 49, 20, 55, 62, 53, 56, 97, 58, 115, 100, 27, 26, 37, 48, 69, 42, 113, 76, 73, 30, 79, 88, 33, 122, 319, 36, 41, 274, 39, 64, 121, 86, 185

Offset: 0

Antti Karttunen, Jan 03 2004. Name changed Aug 16 2014

nonn

This "deeply multiplicative" isomorphism is one of the deep variants of A091202 which satisfies most of the same identities as the latter, but it additionally preserves also the structures where we recurse on prime's index. E.g. we have: A091230(n) = a(A007097(n)) and A061775(n) = A091238(a(n)). This is because the permutation induces itself when it is restricted to the primes: a(n) = A091227(a(A000040(n))).

On the other hand, when this permutation is restricted to the nonprime numbers (A018252), permutation A245814 is induced.

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091205.
Similar or related permutations: A091202, A106442, A106444, A106446, A235041, A245703, A245814, A245822.
Cf. A000040, A007097, A010051, A014580, A018252, A048720, A061775, A091225, A091230, A091238, A091242.

v014580 = vector(2^18); A014580(n) = v014580[n];
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
for(n=0, 8192, write("b091204.txt", n, " ", A091204(n)));
\\ Antti Karttunen, Aug 16 2014

a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes with index i and for composites a(p_i * p_j * ...) = a(p_i) X a(p_j) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720).
As a composition of related permutations:
a(n) = A245703(A245822(n)).
Other identities.
For all n >= 0, the following holds:
a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A245703 has the same property]
For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes bijectively to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242, in some order. The permutations A091202, A106442, A106444, A106446, A235041 and A245703 have the same property.]

A106350 Semiprimes indexed by primes.

Original entry on oeis.org

6, 9, 14, 21, 33, 35, 49, 55, 65, 86, 91, 115, 122, 129, 142, 159, 183, 187, 206, 215, 218, 247, 259, 287, 303, 319, 323, 334, 339, 358, 403, 415, 446, 451, 482, 489, 511, 527, 537, 553, 573, 581, 626, 633, 655, 667, 698, 737, 753, 758, 771, 791, 794, 835, 851

Offset: 1

Jonathan Vos Post, Apr 30 2005

This is the sequence of the n-th semiprime for n = {2,3,5,7,11,13,17,19,23,29...}. Not to be confused with A106349: Primes indexed by semiprimes. We seek to know what this sequence is asymptotically, as J. B. Rosser's result, subsequently modified, is that prime(n) ~ n*(log n + log log n - 1). hence semiprime(prime(n)) ~ semiprime(n)*(log semiprime(n) + log log semiprime(n) - 1). But what is, asymptotically, semiprime(n)?

Semiprime(n) ~ n log n / log log n, hence a(n) ~ n log^2 n / log log n. - Charles R Greathouse IV, Dec 28 2011

			a(1) = semiprime(prime(1)) = semiprime(2) = 6.
a(2) = semiprime(prime(2)) = semiprime(3) = 9.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
J. B. Rosser, The n-th Prime is Greater than n log(n), Proc. London Math. Soc. 45, 21-44, 1939.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A000040, A001358, A007097, A091022, A105997, A105998, A106349.

A001358 := proc(n) if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a ; end if; end do ; end if ; end proc: A106350 := proc(n) A001358(ithprime(n)) ; end proc: seq(A106350(n),n=1..80) ; # R. J. Mathar, Dec 14 2009

terms = 55;
semiPrimes = Select[Range[16 terms], PrimeOmega[#] == 2&];
(* NB If the index Prime[terms] exceeds the size of the table semiPrimes, then the coefficient 16 has to be increased according to the number of terms desired: for instance, for 1000 terms, replace 16 with 32. *)
a[n_] := semiPrimes[[Prime[n]]];
Array[a, terms] (* Jean-François Alcover, Apr 13 2020 *)

a(n) = semiprime(prime(n)). a(n) = A001358(A000040(n)).

a(n) ~ n log^2 n / log log n. - Charles R Greathouse IV, Dec 28 2011

All values after a(32) corrected by R. J. Mathar, Dec 14 2009

A279065 Fermi-Dirac primeth recurrence: a(0)=1; thereafter a(n+1) = a(n)-th number of the form p^(2^k) where p is prime and k>=0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 19, 47, 169, 907, 6829, 67931, 851891, 13034887, 237522877, 5057212439, 123890683831

Offset: 0

Gus Wiseman, Dec 10 2016

Daniel Forgues (see A182979) and Reinhard Zumkeller (see A213925) describe the increasing sequence of positive integers of the form p^(2^k) where p is prime and k>=0 (A050376 or A084400) as Fermi-Dirac primes, because any positive integer has a unique factorization into distinct terms.

OEIS Wiki, "Fermi-Dirac representation" of n.

Cf. A007097, A050376, A084400, A182979, A213925.

nn=10000;FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];
NestWhileList[Part[FDprimeList,#]&,1,#<=Length[FDprimeList]&]

lista(kmax) = {my(m = 1, c=0, isp); print1(1, ", "); for(k = 1, kmax, isp = isprimepower(k); if(isp && isp >> valuation(isp, 2) == 1, c++); if(c == m, print1(k,", "); m=k));} \\ Amiram Eldar, Oct 05 2023

a(15)-a(17) from Amiram Eldar, Oct 05 2023

A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 21, 24, 26, 27, 28, 32, 36, 38, 39, 42, 46, 48, 49, 52, 54, 56, 57, 63, 64, 69, 72, 74, 76, 78, 81, 84, 86, 91, 92, 96, 98, 104, 106, 108, 111, 112, 114, 117, 122, 126, 128, 129, 133, 138, 144, 146, 147, 148, 152, 156, 159

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one, two, and all nonprime numbers whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.

			The sequence of all semi-lone-child-avoiding rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              46: {1,9}             98: {1,4,4}
    2: {1}             48: {1,1,1,1,2}      104: {1,1,1,6}
    4: {1,1}           49: {4,4}            106: {1,16}
    6: {1,2}           52: {1,1,6}          108: {1,1,2,2,2}
    8: {1,1,1}         54: {1,2,2,2}        111: {2,12}
    9: {2,2}           56: {1,1,1,4}        112: {1,1,1,1,4}
   12: {1,1,2}         57: {2,8}            114: {1,2,8}
   14: {1,4}           63: {2,2,4}          117: {2,2,6}
   16: {1,1,1,1}       64: {1,1,1,1,1,1}    122: {1,18}
   18: {1,2,2}         69: {2,9}            126: {1,2,2,4}
   21: {2,4}           72: {1,1,1,2,2}      128: {1,1,1,1,1,1,1}
   24: {1,1,1,2}       74: {1,12}           129: {2,14}
   26: {1,6}           76: {1,1,8}          133: {4,8}
   27: {2,2,2}         78: {1,2,6}          138: {1,2,9}
   28: {1,1,4}         81: {2,2,2,2}        144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}     84: {1,1,2,4}        146: {1,21}
   36: {1,1,2,2}       86: {1,14}           147: {2,4,4}
   38: {1,8}           91: {4,6}            148: {1,1,12}
   39: {2,6}           92: {1,1,9}          152: {1,1,1,8}
   42: {1,2,4}         96: {1,1,1,1,1,2}    156: {1,1,2,6}

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The enumeration of these trees by leaves is A050381.
The locally disjoint version A331873.
The enumeration of these trees by nodes is A331934.
The case with at most one distinct non-leaf branch of any vertex is A331936.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000081, A000669, A007097, A061775, A196050, A320269, A330951, A331871, A331872, A331874, A331937, A331965.

mseQ[n_]:=n==1||n==2||!PrimeQ[n]&&And@@mseQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],mseQ]

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A292050 Matula-Goebel numbers of semi-binary rooted trees.

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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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A306202 Matula-Goebel numbers of rooted semi-identity trees.

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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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A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

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A106350 Semiprimes indexed by primes.

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