A079254 -id:A079254 - cp's OEIS frontend

A079258 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a square".

0, 1, 3, 4, 9, 10, 11, 12, 13, 16, 25, 36, 49, 64, 65, 66, 81, 82, 83, 84, 85, 86, 87, 88, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153

Offset: 0

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

Also, a(n) is smallest nonnegative integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n^2.

Ivan Neretin, Table of n, a(n) for n = 0..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

See A079000, A079253, A079254, A079256, A079257 for similar sequences.

a = {1, 3}; Do[AppendTo[a, If[MemberQ[a, n], Position[a, n][[1, 1]]^2, a[[-1]] + 1]], {n, 3, 58}]; Prepend[a, 0] (* Ivan Neretin, Jul 09 2015 *)

A079256 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a power of 2".

Original entry on oeis.org

1, 2, 5, 6, 8, 16, 17, 32, 33, 34, 35, 36, 37, 38, 39, 64, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 32769, 32770, 32771, 32772, 32773, 32774, 32775, 32776, 32777, 32778

Offset: 1

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

Robert Israel, Table of n, a(n) for n = 1..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

See A079000, A079253, A079254 for similar sequences.

A:= Vector(100):
A[1..5]:= <1,2,5,6,8>:
going:= true;
for n from 3 while going do
  for k from 0 to A[n+1]-A[n] do
    if A[n]+k > 100 then going:= false; break fi;
    A[A[n]+k]:=  2^n+k
od od:
convert(A,list); # Robert Israel, Mar 20 2019

A085925 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not prime".

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 12, 14, 17, 18, 19, 20, 23, 29, 30, 32, 33, 34, 37, 41, 42, 43, 47, 53, 59, 61, 62, 63, 67, 68, 69, 70, 71, 73, 74, 79, 83, 89, 90, 91, 92, 97, 101, 103, 104, 107, 109, 113, 127, 131, 132, 137, 139, 149, 151, 157, 158, 163, 164, 165, 166, 167, 173

Offset: 1

David Wasserman, Aug 16 2003

			a(4) = 6 because 4 is in the sequence and 6 is the next nonprime after a(3). a(5) = 7 because 5 is not in the sequence and 7 is the next prime after a(4).

Cf. A079000, A079254.

A companion to A121053.

A324854 Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.

Original entry on oeis.org

1, 4, 7, 8, 14, 16, 17, 19, 28, 32, 34, 38, 43, 49, 53, 56, 59, 64, 67, 68, 76, 86, 98, 106, 107, 112, 118, 119, 128, 131, 133, 134, 136, 139, 152, 163, 172, 191, 196, 212, 214, 224, 227, 236, 238, 241, 256, 262, 263, 266, 268, 272, 277, 278, 289, 301, 304

Offset: 1

Gus Wiseman, Mar 18 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.

A multiplicative semigroup: if x and y are in the sequence then so is x*y. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   4: {1,1}
   7: {4}
   8: {1,1,1}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  38: {1,8}
  43: {14}
  49: {4,4}
  53: {16}
  56: {1,1,1,4}
  59: {17}
  64: {1,1,1,1,1,1}
  67: {19}
  68: {1,1,7}

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, The rooted trees whose Matula-Goebel numbers are the first 64 terms.

Cf. A000002, A000720, A001462, A079254, A112798, A276625, A290822, A304360.

Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324855.

S:= {1}:
for n from 3 to 400 do
  if map(numtheory:-pi, numtheory:-factorset(n)) subset S then
    S:= S union {n}
  fi
od:
sort(convert(S,list)); # Robert Israel, Mar 19 2019

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

A379051 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is composite.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 12, 14, 15, 7, 16, 17, 18, 20, 21, 22, 24, 23, 25, 26, 27, 28, 30, 32, 33, 34, 35, 31, 36, 38, 39, 40, 42, 44, 45, 41, 46, 48, 49, 50, 51, 47, 52, 54, 55, 56, 57, 58, 60, 62, 63, 59, 64, 65, 66, 68, 69, 70, 72, 67, 74, 75, 76, 77, 78, 80

Offset: 1

N. J. A. Sloane, Dec 17 2024

nonn

The sequence tells you exactly which terms of the sequence are composite: the second, fourth, fifth, sixth, etc. terms are composite, and this is the lexicographically earliest sequence with this property.
Let P be a property of the nonnegative integers, such as being a prime.
The OEIS contains many entries whose definitions have the following form.
"The sequence is the lexicographically earliest infinite sequence of distinct positive (or sometimes nonnegative) integers with the property that n is a term of the sequence iff a(n) has property P."
That is, the terms of the sequence tell you which terms of the sequence have the property. A121053 is the classical example.
Since these are lists, the offset is usually 1.
There are two versions, one where the sequence is required to be strictly increasing, and an unrestricted version which is not required to be increasing.
Examples:
  Property P    Unrestricted    Increasing
  ----------------------------------------
  Prime         A121053         A079254, A334067 (offset 0)
  Even          A080032         A079253
  Odd           A079313         A079000
  Composite     A379051         A099797
  Not composite A377901         A099798
  Not prime     A379053         A085925

Michael De Vlieger, Table of n, a(n) for n = 1..65536 [Terms 1 to 10000 from Scott R. Shannon]

Cf. A079000, A079253, A079254, A079313, A080032, A085925, A099797, A099798, A121053, A334067, A377901, A379053.

A379053 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is not a prime.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 12, 14, 13, 15, 16, 18, 20, 21, 19, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 23, 36, 37, 38, 39, 40, 42, 44, 45, 46, 48, 49, 43, 50, 51, 52, 54, 55, 53, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 61, 70, 72, 74, 75, 76, 77, 78, 71

Offset: 1

N. J. A. Sloane, Dec 17 2024

nonn

The sequence tells you exactly which terms of the sequence are either 1 or composite.

See the Comments in A379051 for further information.

Michael De Vlieger, Table of n, a(n) for n = 1..65536 [Terms 1 to 10000 from Scott R. Shannon]

Cf. A079000, A079253, A079254, A079313, A080032, A085925, A099797, A099798, A121053, A334067, A377901, A379051.

nn = 120; u = 3; v = {}; w = {}; c = 4;
{1}~Join~Reap[Do[
  If[MemberQ[w, n],
    k = c; w = DeleteCases[w, n],
    m = Min[{c, u, v}]; If[And[CompositeQ[m], n < m],
      AppendTo[v, n]];
      If[Length[v] > 0,
        If[v[[1]] == m,
        v = Rest[v]]]; k = m];
    AppendTo[w, k]; If[k == c, c++; While[PrimeQ[c], c++]]; Sow[k];
If[n + 1 >= u, u++; While[CompositeQ[u], u++]], {n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Dec 17 2024 *)

When sorted, this appears to be the complement of [2, 5, 11, 17, 29, and prime(2*t+1), t >= 35]. - Scott R. Shannon, Dec 18 2024

More terms from Michael De Vlieger, Dec 17 2024

A079257 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a triangular number".

Original entry on oeis.org

0, 1, 4, 5, 6, 10, 15, 16, 17, 18, 21, 22, 23, 24, 25, 28, 36, 45, 55, 56, 57, 66, 78, 91, 105, 120, 121, 122, 136, 137, 138, 139, 140, 141, 142, 143, 153, 154, 155, 156, 157, 158, 159, 160, 161, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 190, 210, 231

Offset: 0

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

See A079000, A079253, A079254, A079256, A079258 for similar sequences.

A099797 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is composite".

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 11, 12, 14, 17, 18, 20, 23, 24, 29, 31, 32, 33, 37, 38, 41, 43, 44, 45, 47, 53, 59, 61, 62, 67, 68, 69, 70, 71, 73, 79, 80, 81, 83, 89, 90, 97, 98, 99, 100, 101, 102, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 152, 157, 158, 159, 163, 167

Offset: 1

Ray Chandler, Nov 02 2004

nonn

			a(1) cannot be 1 because 1 is not composite; it can be 2.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079254, A085925, A099798.

A099798 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not composite".

Original entry on oeis.org

1, 2, 3, 6, 8, 11, 12, 13, 14, 15, 17, 19, 23, 29, 31, 32, 37, 38, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 59, 61, 62, 63, 64, 65, 67, 71, 72, 74, 79, 83, 84, 89, 97, 98, 101, 103, 107, 109, 113, 127, 131, 137, 138, 140, 141, 142, 149, 150, 151, 157, 163, 167

Offset: 1

Ray Chandler, Nov 02 2004

nonn

			a(4) cannot be 4 because 4 is composite; it cannot be 5, for then 4 is not in the sequence while a(4) is not composite; but 6 is possible.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079254, A085925, A099797.

A317964 Prime numbers in the lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence (A304360).

Original entry on oeis.org

2, 5, 13, 17, 23, 31, 37, 43, 47, 61, 67, 73, 79, 89, 103, 107, 109, 113, 137, 149, 151, 163, 167, 179, 181, 193, 197, 223, 227, 233, 241, 251, 257, 263, 269, 271, 277, 281, 307, 317, 347, 349, 353, 359, 379, 383, 389, 397, 419, 421, 431, 433, 449, 457, 463, 467, 487, 499, 503, 509, 521, 523, 547

Offset: 1

N. J. A. Sloane, Aug 26 2018

nonn

Also primes whose prime index is not in A304360, or is in A324696. A prime index of n is a number m such that prime(m) divides n. - Gus Wiseman, Mar 19 2019

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

Cf. A000720, A001462, A007097, A060197, A079254, A112798, A276625, A277098, A290822, A304360, A306844.

Cf. A324695, A324698, A324704, A324741, A324756, A324758, A324765, A324766, A324768.

count:= 0:
P:= {}: A:= NULL:
for n from 2 while count < 100 do
  pn:= numtheory:-factorset(n);
  if pn intersect P = {} then
    P:= P union {ithprime(n)};
    if isprime(n) then A:= A, n; count:= count+1 fi;
  fi
od:
A; # Robert Israel, Aug 26 2018

aQ[n_]:=n==1||Or@@Cases[FactorInteger[n],{p_,_}:>!aQ[PrimePi[p]]];
Prime[Select[Range[100],aQ]] (* Gus Wiseman, Mar 19 2019 *)

cp's OEIS Frontend

A079258 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a square".

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A079256 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a power of 2".

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A085925 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not prime".

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A324854 Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.

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Maple

Mathematica

A379051 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is composite.

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A379053 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is not a prime.

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A079257 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a triangular number".

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A099797 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is composite".

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A099798 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not composite".

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A317964 Prime numbers in the lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence (A304360).

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