A363016 a(n) is the least integer k such that the k-th, (k+1)-th, ..., (k+n-1)-th primes are congruent to 1 mod 4.

3, 6, 24, 77, 378, 1395, 1395, 1395, 1395, 31798, 61457, 240748, 800583, 804584, 804584, 804584, 16118548, 16138563, 16138563, 56307979, 56307979, 56307979, 56307979, 56307979, 3511121443, 3511121443, 26284355567, 26284355567, 26284355567, 118027458557, 118027458557, 118027458557, 118027458557

Offset: 1

Author

Léo Gratien, May 13 2023

Keywords

• nonn

Comments

a(n) is also the minimal rank where n consecutive 1's appear in A023512.

The sequence is infinite by Shiu's theorem.

Examples

			For n=3, a(3) = 24 because prime(24)+1=90, prime(25)+1=98, and prime(26)+1=102 are the first 3 consecutive primes p such that p+1 is divisible by 2 and not by 4.
		

Links

• D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373.

Crossrefs

Cf. A000720, A057624, A023512.

Cf. A363017 (3 mod 8).

Formula

a(n) = A000720(A057624(n)). - Amiram Eldar, May 13 2023