A147972 Smallest prime p modulo which the first n primes are nonzero quadratic residues.

7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 366791, 366791, 3818929, 9257329, 22000801, 36415991, 48473881, 120293879, 120293879, 131486759, 131486759, 2929911599, 2929911599, 7979490791, 23616331489, 23616331489, 89206899239, 121560956039, 196265095009, 196265095009, 513928659191, 5528920734431, 8402847753431, 8402847753431, 8402847753431, 70864718555231

Offset

1,1

Comments

The same primes without repetitions are listed in A147970.

a(n) <= min{A002223(n), A002224(n)}. What is the smallest n for which this inequality is strict?

By definition, a(n) == 1, 7 (mod 8), so a(n) = min{A002223(n), A002224(n)}. - Jianing Song, Feb 18 2019

Links

Table of n, a(n) for n=1..37.

Formula

a(n) >= min{A002189(n-1), A045535(n-1)}. - Jianing Song, Feb 18 2019

Mathematica

(*version 7.0*)m=1; P=7; Lst={p}; While[m<25, m++; S=Prime[Range[m]]; While[MemberQ[JacobiSymbol[S, p], -1], p=NextPrime[p]]; Lst=Append[Lst, P]]; Lst (* Emmanuel Vantieghem, Jan 31 2012 *)

Prog

(PARI) t=2; forprime(p=2, 1e9, forprime(q=2, t, if(kronecker(q, p)<1, next(2))); print1(p", "); t=nextprime(t+1); p--) \\ Charles R Greathouse IV, Jan 31 2012

Crossrefs

Cf. A000229, A002189, A002223, A002224, A045535, A053760, A133435, A147969, A147970, A147971.

Smallest prime p such that each of the first n primes has q q-th roots mod p: this sequence (q=2), A002225 (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).

Sequence in context: A198644 A045535 A001984 * A002223 A034563 A242496

Adjacent sequences: A147969 A147970 A147971 * A147973 A147974 A147975

Keyword

nonn

Author

Max Alekseyev, Nov 18 2008

Extensions

a(23)-a(25) from Emmanuel Vantieghem, Jan 31 2012

a(26)-a(37) from Max Alekseyev, Aug 21 2015

Status

approved