A386978 - OEIS

A386978

Numbers k such that the k-th prime gap contains an integer whose least prime factor is greater than or equal to the length of the prime gap.

2, 3, 5, 7, 10, 13, 15, 17, 20, 21, 26, 28, 32, 33, 34, 35, 37, 39, 41, 42, 43, 45, 47, 49, 52, 53, 54, 55, 57, 60, 61, 64, 66, 68, 69, 72, 73, 74, 77, 79, 81, 83, 84, 87, 89, 92, 94, 98, 99, 101, 102, 104, 106, 107, 109, 111, 113, 114, 116, 118, 120, 121, 123

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OFFSET

1,1

COMMENTS

If p and q are twin primes and p<q, then the interval (p, q) contains an integer whose least prime factor is greater than or equal to the length of the prime gap (p, q): indeed, the least prime factor of p+1 is greater than or equal to 2 whereas the length of the gap is q-p=2.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Thomas Bloom, Problem 682, Erdős Problems.

Erdős problems database contributors, Erdős problem database, see no. 682.

Ayla Gafni and Terence Tao, Rough numbers between consecutive primes, arXiv:2508.06463 [math.NT] (2025).

EXAMPLE

1 does not belong to this sequence because the first prime gap is (2,3). The second prime gap is (3,5); since the length of this interval is 5-3=2 and 4 belongs to it, we have that 2 is the first term of the sequence.

MAPLE

q:= k-> ((p, r)-> ormap(f-> min(ifactors(f)[2][..., 1])>=

r-p, [$p+1..r-1]))((map(ithprime, [k, k+1])[])):

select(q, [$1..123])[]; # Alois P. Heinz, Aug 12 2025

MATHEMATICA

For[n = 1, n <= 200, n++, k = Prime[n];

count = 0;

While[k < Prime[n+1], If[FactorInteger[k][[1, 1]] < Prime[n+1] - Prime[n], count = count+1]; k++; ];

If[count == Prime[n+1] - Prime[n] - 1, , Print[n]]]

(* This code outputs all the terms of the sequence in the interval [1, 200]. *)

CROSSREFS

Cf. A001223.