Léo Gratien - cp's OEIS frontend

User: Léo Gratien

Léo Gratien has authored 2 sequences.

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

Léo Gratien, May 13 2023

nonn

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

The sequence is infinite by Shiu's theorem.

			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.

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

Cf. A000720, A057624, A023512.

Cf. A363017 (3 mod 8).

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

A363017 a(n) is the least integer k such that the k-th, (k+1)-th, ..., (k+n-1)-th primes are congruent to 3 mod 8.

Original entry on oeis.org

2, 94, 334, 4422, 23969, 303493, 303493, 606529, 28725046, 92865581, 397316305, 511883558, 848516256, 23738949809, 144899085865, 469694200388, 3800553021301, 8571139291304, 63858322306341, 90990757864814

Offset: 1

Léo Gratien, May 13 2023

a(n) is also the minimal rank where n consecutive 2's appear in A023512.

The sequence is infinite by Shiu's theorem.

			For n=2, a(2) = 94 because prime(94)+1 = 492 = 4*123, prime(95)+1 = 500 = 4*125 are the first two consecutive primes p such that p+1 is divisible by 4 and not by 8.

Cf. A057632, A023512.

Cf. A363016 (with 1 mod 4).

a(n) = primepi(A057632(n)). - Amiram Eldar, May 13 2023

a(19)-a(20) from Martin Ehrenstein, May 28 2023

cp's OEIS Frontend

User: Léo Gratien

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.

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