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A333924 Smallest prime of the form 4*k + 3 that is a divisor of 4*n! - 1. 0
3, 3, 7, 23, 19, 479, 2879, 19, 179, 2551, 14515199, 159667199, 26246663, 47, 3007159, 85303, 43, 455999, 13099, 311369011223, 7791519641878751, 59, 50207, 149709500816123, 71, 61651424911, 1146111319366855507, 3902575987, 27963070149883187169101323, 3262754470190705587633531 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Every integer equal to 4*n!-1 (A173321) has a prime factor > n of the form 4*k+3; this is one of the proofs which show that there are infinitely many primes of the form 4*k+3 (A002145).

LINKS

Table of n, a(n) for n=0..29.

EXAMPLE

4*11!-1 = 159667199 that is prime of the form 4*k+3, hence a(11) = 159667199.

4*13!-1 = 24908083199 = 47 * 2963 * 178859, these 3 primes factors are all of the form 4*k+3, the smallest one is 47 hence a(13) = 47.

4*14!-1 = 348713164799 = 61 * 1901 * 3007159, only 3007159 is a prime of the form 4*k+3, hence a(14) = 3007159.

MATHEMATICA

a[n_] := Min[Select[First /@ FactorInteger[4*n! - 1], Mod[#, 4] == 3 &]]; Array[a, 30, 0] (* Amiram Eldar, Apr 10 2020 *)

PROG

(PARI) a(n) = {my(f=factor(4*n!-1)[, 1]); for(i=1, #f, if(f[i]%4==3, return(f[i]))); } \\ Jinyuan Wang, Apr 10 2020

CROSSREFS

Cf. A002144, A173321.

Subsequence of A002145.

Sequence in context: A100666 A262375 A232368 * A173321 A191498 A065747

Adjacent sequences:  A333921 A333922 A333923 * A333925 A333926 A333927

KEYWORD

nonn

AUTHOR

Bernard Schott, Apr 10 2020

EXTENSIONS

a(23) corrected by and more terms from Jinyuan Wang, Apr 10 2020

STATUS

approved

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Last modified May 17 00:16 EDT 2021. Contains 343957 sequences. (Running on oeis4.)