A391354 Numbers k such that 4*k - (greatest prime < 4*k) < (least prime > 4*k) - 4*k.

2, 5, 6, 8, 11, 12, 17, 20, 21, 23, 26, 29, 32, 33, 35, 38, 41, 42, 46, 50, 51, 53, 54, 56, 61, 63, 66, 68, 71, 74, 77, 80, 83, 85, 90, 92, 95, 96, 98, 101, 103, 106, 110, 111, 113, 116, 117, 118, 120, 122, 123, 125, 126, 128, 131, 132, 137, 141, 143, 145

Offset

1,1

Comments

Let (f(k)) be an increasing sequence of positive composite numbers. Let u(k) = greatest prime < f(k) and v(k) = least prime > f(k). Let

s(1) = {k : f(k) - u(k) < v(k) - f(k)} = {k : f(k) < (u(k)+v(k))/2};

s(2) = {k : f(k) - u(k) = v(k) - f(k)} = {k : f(k) = (u(k)+v(k))/2};

s(3) = {k : f(k) - u(k) > v(k) - f(k)} = {k : f(k) > (u(k)+v(k))/2}.

The sets s(1), s(2), s(3) partition the natural numbers, A000027.

See A390788 for a guide to related sequences.

The 3-way partition of A000027:

this sequence: (2, 5, 6, 8, 11, 12, 17, 20, 21, 23, 26, 29, 32, 33, 35, 38, 41, 42, ...)

A391355: (1, 3, 14, 15, 16, 18, 19, 27, 30, 36, 40, 44, 45, 48, 57, 59, 60, 65, 72, ...)

A391356: (4, 7, 9, 10, 13, 22, 24, 25, 28, 31, 34, 37, 39, 43, 47, 49, 52, 55, 58, ...)

The primes indexed by s(1), s(2), s(3) are partitioned into three sequences as follows:

Prime(s(1)) = (3, 11, 13, 19, 31, 37, 59, 71, 73, 83, 101, 109, 131, 137, 149, ...)

Prime(s(2)) = (2, 5, 43, 47, 53, 61, 67, 103, 113, 151, 173, 193, 197, 223, ...)

Prime(s(3)) = (7, 17, 23, 29, 41, 79, 89, 97, 107, 127, 139, 157, 167, 191, ...)

Links

Table of n, a(n) for n=1..60.

Mathematica

m = 4; z = 140;
u[n_] := NextPrime[m n, -1]; v[n_] := NextPrime[m n];
Select[Range[400], m*# - v[#] < u[#] - m*# &]    (* A391354 *)
Select[Range[400], m*# - v[#] == u[#] - m*# &]   (* A391355 *)
Select[Range[400], m*# - v[#] > u[#] - m*# &]    (* A391356 *)

Crossrefs

Cf. A000027, A000040, A390788, A391355, A391356.

Sequence in context: A047267 A058591 A247062 * A059009 A214642 A026179

Adjacent sequences: A391351 A391352 A391353 * A391355 A391356 A391357

Keyword

nonn

Author

Clark Kimberling, Dec 14 2025

Status

approved