A147971 -id:A147971 - cp's OEIS frontend

A088192 Distance between prime(n) and the largest quadratic residue modulo prime(n).

1, 2, 1, 3, 2, 1, 1, 2, 5, 1, 3, 1, 1, 2, 5, 1, 2, 1, 2, 7, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 2, 5, 1, 2, 1, 7, 1, 1, 3, 2, 3, 2, 1, 1, 7, 1, 2, 1, 5, 1, 3, 1, 1, 2, 1, 2, 11, 1, 1, 2, 1, 2, 1, 1, 7, 3, 1, 2, 5, 1, 1, 1, 1, 2, 1, 7, 1, 3, 2, 1, 1, 1, 3, 2, 13, 3, 2, 2, 5, 1, 1, 2, 1

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

a(n) = smallest m>0 such that -m is a quadratic residue modulo prime(n).
a(n) = smallest m>0 such that prime(n) either splits or ramifies in the imaginary quadratic field Q(sqrt(-m)). Equals -A220862(n) except when n = 1. Cf. A220861, A220863. - N. J. A. Sloane, Dec 26 2012
The values are 1 or a prime number (easily provable!). The maximum occurring prime values increase very slowly: up to 10^5 terms the largest prime is 43. The primes do not appear in order.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Cor. 5.17, p. 105. - From N. J. A. Sloane, Dec 26 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ferenc Adorjan, The sequence of largest quadratic residues modulo the primes.
J. A. Bergstra, I. Bethke, A negative result on algebraic specifications of the meadow of rational numbers, arXiv preprint arXiv:1507.00548 [math.RA], 2015-2016.

Records are (essentially) given by A147971.

Cf. A088190, A088191, A088193, A088194, A088195, A220861, A220862, A220863.

a[n_] := With[{p = Prime[n]}, If[JacobiSymbol[-1, p] > 0, 1, For[d = 2, True, d = NextPrime[d], If[JacobiSymbol[-d, p] >= 0, Return[d]]]]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018, after Charles R Greathouse IV *)

qrp_pm(fr,to)= {/* The distance of largest QR modulo the primes from the primes */ local(m,p,v=[]); for(i=fr,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

do(p)=if(kronecker(-1,p)>0, 1, forprime(d=2, p, if(kronecker(-d, p) >= 0, return(d))))
apply(do, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

a(n) = A053760(n) unless -1 is a quadratic residue mod prime(n). - Charles R Greathouse IV, Oct 31 2012

Edited by Max Alekseyev, Oct 29 2012

A147969 Smallest prime p modulo which numbers 1,2,...,n are quadratic residues.

Original entry on oeis.org

2, 7, 23, 23, 71, 71, 311, 311, 311, 311, 479, 479, 1559, 1559, 1559, 1559, 5711, 5711, 10559, 10559, 10559, 10559, 18191, 18191, 18191, 18191, 18191, 18191, 31391, 31391, 366791, 366791, 366791, 366791, 366791, 366791, 366791, 366791, 366791

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

The same primes without repetitions are listed in A147970.

Charles R Greathouse IV, Table of n, a(n) for n = 1..100

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

a(n)=forprime(p=2,default(primelimit),forprime(i=2,n, if(kronecker(i,p)<1,next(2)));return(p)) \\ Charles R Greathouse IV, Apr 06 2012

A147970 Primes corresponding to the records in the sequence of smallest positive quadratic nonresidues (A053760).

Original entry on oeis.org

2, 7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 3818929, 9257329, 22000801, 36415991, 48473881, 120293879, 131486759, 2929911599, 7979490791, 23616331489, 89206899239, 121560956039, 196265095009, 513928659191, 5528920734431, 8402847753431, 70864718555231

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

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

Prime p=A000040(n) is in this sequence iff A053760(k) < A053760(n) for every kA000040(A147971(n))

a(20)-a(29) from Charles R Greathouse IV, Apr 06 2012

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

Original entry on oeis.org

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

Max Alekseyev, Nov 18 2008

nonn

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

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).

(*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 *)

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

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

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

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

A210488 a(n) is the index of A210487 when the n-th prime appears for the first time.

Original entry on oeis.org

3, 4, 2, 20, 60, 92, 246, 752, 1289, 2084, 3383

Offset: 1

Lei Zhou, Jan 23 2013

			In A210487, 2 appears for the first time as A210487(3), so a(1)=3;
3 appears for the first time as A210487(4), so a(2)=4;
5 appears for the first time as A210487(2), so a(3)=2.

Cf. A000040, A210487, A147971.

N := 60 ;
A210488 := [seq(0,i=1..N)] ;
for n from 2 do
    a := A210487(n) ;
    if isprime(a) then
        idx := numtheory[pi](a) ;
        if idx <= N then
        if op(idx,A210488) = 0 then
            A210488  := subsop(idx=n,A210488) ;
            print(A210488) ;
        end if;
        end if;
    end if;
end do: # R. J. Mathar, Apr 17 2013

nn = 11; a = Table[0, {nn}]; k = 1; While[Times @@ a == 0, k++; p1 = Prime[k]; id = PrimePi[Min[Table[Last[FactorInteger[Prime[k]^2 - Prime[j]^2]][[1]], {j, k - 1}]]]; If[id <= nn && a[[id]] == 0, a[[id]] = k]]; a

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A088192 Distance between prime(n) and the largest quadratic residue modulo prime(n).

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A147969 Smallest prime p modulo which numbers 1,2,...,n are quadratic residues.

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A147970 Primes corresponding to the records in the sequence of smallest positive quadratic nonresidues (A053760).

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

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PARI

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A210488 a(n) is the index of A210487 when the n-th prime appears for the first time.

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