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

Offset: 1

Author

José Hernández, Aug 11 2025

Keywords

• nonn
• easy

Comments

If p and q are twin primes and p

Examples

			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.
		

Links

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

Crossrefs

Cf. A001223.

Programs

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]. *)

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