A396279 Integers k such that p - k = floor(log_2(k)), where p is the smallest prime strictly greater than k.

2, 5, 8, 14, 19, 25, 32, 48, 54, 62, 73, 83, 91, 121, 142, 184, 204, 216, 244, 285, 299, 323, 339, 359, 389, 401, 411, 423, 449, 471, 479, 491, 512, 532, 548, 578, 622, 632, 664, 692, 710, 778, 788, 800, 812, 830, 844, 868, 898, 920, 958, 1000, 1022, 1039, 1051

Offset

1,1

Links

Table of n, a(n) for n=1..55.

Example

For a(3) = 8: The smallest prime > 8 is 11. The distance to the next prime is 11 - 8 = 3. The base-2 scale is floor(log2(8)) = 3. Because the gap (3) equals the scale (3), 8 is included.
For a(7) = 32: The smallest prime > 32 is 37. The distance to the next prime is 37 - 32 = 5. The base-2 scale is floor(log2(32)) = 5. Because the gap (5) equals the scale (5), 32 is included.

Maple

q:= k-> nextprime(k)-k=ilog2(k):
select(q, [$1..1070])[];  # Alois P. Heinz, May 23 2026

Mathematica

Select[Range[1100], NextPrime[#] == # + Floor[Log2[#]] &] (* Amiram Eldar, May 22 2026 *)

Prog

(Python)
import math
import sympy
def generate_boundary_numbers(limit):
    # These lines must be indented by 4 spaces
    boundary_numbers = []
    n = 2
    while len(boundary_numbers) < limit:
        # These lines are inside 'while', so they need 8 spaces
        next_p = sympy.nextprime(n)
        gap = next_p - n
        scale = int(math.log2(n))
        if gap == scale:
            # This line is inside 'if', so it needs 12 spaces
            boundary_numbers.append(n)
        n += 1
    return boundary_numbers
# This is outside the function, so it has 0 spaces
print(generate_boundary_numbers(50))
(PARI) isok(k) = nextprime(k+1) - k == log(k)\log(2); \\ Michel Marcus, May 21 2026

Crossrefs

Cf. A000040 (primes), A000523 (floor(log2)), A151800 (next prime), A396203, A396282.

Sequence in context: A190105 A295400 A266287 * A111711 A095348 A215725

Adjacent sequences: A396276 A396277 A396278 * A396281 A396282 A396283

Keyword

nonn

Author

Sajid Hussain, May 21 2026

Status

approved