A379051 - OEIS

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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

When sorted, this appears to be the complement of [1, 3, 11, and prime(2*t), t >= 3]. - Scott R. Shannon, Dec 18 2024

MATHEMATICA

nn = 120; u = 3; v = {}; w = {2}; c = 4;

{2}~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 *)

CROSSREFS

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