cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A057632 Initial prime in first sequence of n primes congruent to 3 modulo 8.

Original entry on oeis.org

3, 491, 2243, 42299, 274123, 4310083, 4310083, 9065867, 547580443, 1885434347, 8674616939, 11312238283, 19201563659, 619849118491, 4056100954547, 13721202685691, 119254168189363, 276151474703651, 2189798979924331, 3153425741761723
Offset: 1

Views

Author

Robert G. Wilson v, Oct 10 2000

Keywords

Examples

			a(3) = 2243 because this number is the first in a sequence of 3 consecutive primes all of the form 8*n + 3.

Links

Crossrefs

Cf. A363017 (indices), A057624 (with 1 modulo 4).

Programs

  • Mathematica
    NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; Return[ k ] ]; p = 0; Do[ a = Table[ -1, {n} ]; k = Max[ 1, p ]; While[ Union[ a ] != {3}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 8 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 9} ]

Extensions

More terms from Jens Kruse Andersen, May 28 2006
a(16)-a(18) from Giovanni Resta, Aug 04 2013
a(19)-a(20) from Martin Ehrenstein, May 28 2023

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.

Original entry on oeis.org

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

Views

Author

Léo Gratien, May 13 2023

Keywords

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

Crossrefs

Cf. A000720, A057624, A023512.
Cf. A363017 (3 mod 8).

Formula

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