A386995 Primes p such that 2p-1 is a square and 2p+1 is also prime.

5, 41, 113, 761, 1013, 14621, 23981, 47741, 51521, 59513, 70313, 97241, 185441, 244301, 252761, 335381, 391613, 413141, 481181, 511061, 695021, 859361, 883121, 1278401, 1336613, 1529501, 1746581, 1872113, 2111513, 2161121, 2287661, 2619761, 3289613, 3351461, 3540461

Offset

1,1

Comments

Might be called "Masser primes" after David Masser, who on December 29, 2025 sent me an email in which he found that 2025 is a square, the next year is 2026 = 2 * 1013, 1013 is a prime, and interestingly the year after 2026 is 2027, a prime. He also asked: (a) Find the next time this unlikely intersection occurs. (b) Assuming H prove that it occurs infinitely often. (where H can be a good joke :-))

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

a(n) == 5 (mod 6). - Ivan N. Ianakiev, Jan 05 2026

a(n) == 5 (mod 12). - Robert Israel, Jan 05 2026

a(n) == 5 (mod 36). - Jason Yuen, Jan 05 2026

Example

5 is a term, since 2*5-1 = 9 is a square and 2*5+1 = 11 is a prime.

Maple

q:= p-> andmap(isprime, [p, 2*p+1]):
select(q, [seq((n+1)*2*n+1, n=0..1500)])[];  # Alois P. Heinz, Dec 29 2025

Mathematica

Select[Prime[Range[260000]], IntegerQ[Sqrt[2*#-1]] && PrimeQ[2*#+1] &] (* Amiram Eldar, Dec 29 2025 *)
Select[Prime[Range[260000]], Mod[#, 6]==5&&IntegerQ[Sqrt[2*#-1]]&&PrimeQ[2*#+1]&] (* made a little faster by Ivan N. Ianakiev, Jan 05 2026 *)

Crossrefs

Intersection of A027862 and A005384.

Sequence in context: A139846 A201718 A142101 * A102265 A337393 A262340

Adjacent sequences: A386992 A386993 A386994 * A386996 A386997 A386998

Keyword

nonn

Author

Shaoshi Chen, Dec 29 2025

Status

approved