A092419 -id:A092419 - cp's OEIS frontend

A067073 Records in A092419.

3, 7, 13, 17, 19, 31, 37, 41, 43, 47, 53, 71, 79, 89, 103, 127, 131, 137, 151

Offset: 1

N. J. A. Sloane, Oct 16 2008

The old entry with this sequence number was a duplicate of A049236.

A000037 := proc(n) n +floor(1/2+sqrt(n)) ; end: A092419 :=proc(n) local i, k; k := A000037(n) ; for i from 1 do if numtheory[legendre](k, ithprime(i)) = -1 then RETURN(ithprime(i)); fi; od; end: rec := -1 ; for n from 1 do a := A092419(n) ; if a > rec then printf("%d,\n",a) ; rec := a; fi; od: # R. J. Mathar, Jul 13 2009

a(8)-a(12) from R. J. Mathar, Jul 13 2009

a(13)-a(19) from Sean A. Irvine, Dec 02 2023

A070040 Where records occur in A092419.

Original entry on oeis.org

1, 4, 12, 104, 231, 380, 3039, 12070, 22609, 47955, 60409, 182492, 859647, 8039644, 11990003, 78269808, 6091154409, 6941215856, 13803257219

Offset: 1

N. J. A. Sloane, Oct 16 2008

The old entry with this sequence number was a duplicate of A056924.

Cf. A092419, A067073.

A092419(a(n))=A067073(n). [From R. J. Mathar, Jul 13 2009]

a(8)-a(12) from R. J. Mathar, Jul 13 2009

a(13)-a(19) from Sean A. Irvine, Dec 02 2023

A232931 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).

Original entry on oeis.org

2, 3, 5, 2, 3, 2, 7, 5, 2, 5, 2, 7, 3, 3, 2, 3, 5, 13, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 5, 2, 3, 11, 2, 3, 11, 7, 7, 2, 7, 3, 3, 2, 7, 2, 3, 11, 2, 3, 2, 5, 5, 2, 5, 2, 11, 3, 3, 5, 2, 7, 11, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 7, 3, 11, 2, 3, 7, 7, 5, 2, 5, 2, 13, 3, 3, 2, 2, 3, 2, 3, 2, 5, 5, 11, 2, 7, 5, 3, 3, 5, 2, 3, 13, 5, 2, 3, 2, 17, 2, 2, 7, 3, 3, 2, 13, 2, 5, 2, 3, 5, 7, 5, 2, 5, 2, 11, 3, 2, 5, 2, 3, 7, 2, 3, 2, 17, 5, 7, 2, 7, 2, 5, 3, 3, 7, 2, 3, 7, 5, 2, 3

Offset: 2

Steven Finch, Dec 02 2013

nonn

From Jianing Song, Jan 30 2019: (Start)
a(n) is necessarily prime. Otherwise, if a(n) is not prime, we have (D/p) = 0 or 1 for all prime divisors p of a(n), so (D/a(n)) must be 0 or 1 too, a contradiction.
a(n) is the least inert prime in the real quadratic field with discriminant D, D = A003658(n). (End)

			A003658(3) = 8, (8/3) = -1 and (8/2) = 0, so a(3) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Andrew Granville, R. A. Mollin and H. C. Williams, An upper bound on the least inert prime in a real quadratic field, Canad. J. Math. 52:2 (2000), pp. 369-380.
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 1185-1202.
Enrique Treviño, The least inert prime in a real quadratic field, Mathematics of Computation 81:279 (2012), pp. 1777-1797. See also his PANTS 2010 talk.

Cf. A003658, A092419, A241482.

nMax = 200; A003658 = Select[Range[4nMax], NumberFieldDiscriminant[Sqrt[#]] == #&]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k]]]; a[n_] := f[A003658[[n]]]; Table[a[n], {n, 2, nMax}] (* Jean-François Alcover, Nov 05 2016 *)

lp(D)=forprime(p=2,,if(kronecker(D,p)<0,return(p)))
for(n=5,1e3,if(isfundamental(n),print1(lp(n)", "))) \\ Charles R Greathouse IV, Apr 23 2014

With D = A003658(n): Mollin conjectured, and Granville, Mollin, & Williams proved, that for n > 1128, a(n) <= D^0.5 / 2. Treviño proves that for n > 484, a(n) <= D^0.45. Asymptotically the best known upper bound for the exponent is less than 0.16 when D is prime and 1/4 + epsilon (for any epsilon > 0) for general D. - Charles R Greathouse IV, Apr 23 2014 (corrected by Enrique Treviño, Mar 18 2022)

a(n) = A092419(A003658(n) - floor(sqrt(A003658(n)))), n >= 2. - Jianing Song, Jan 30 2019

Name simplified by Jianing Song, Jan 30 2019

A144294 Let k = n-th nonsquare = A000037(n); then a(n) = smallest prime p such that k is not a square mod p.

Original entry on oeis.org

3, 5, 3, 7, 5, 3, 7, 3, 5, 5, 3, 13, 3, 5, 7, 3, 11, 5, 3, 7, 3, 5, 5, 3, 11, 7, 3, 5, 7, 3, 5, 3, 11, 7, 3, 5, 5, 3, 7, 11, 3, 5, 3, 11, 5, 3, 7, 7, 3, 5, 5, 3, 13, 7, 3, 5, 3, 7, 5, 3, 7, 13, 3, 5, 5, 3, 7, 7, 3, 5, 11, 3, 5, 3, 11, 11, 3, 5, 5, 3, 7, 17, 3, 5, 7, 3, 7, 5, 3, 13

Offset: 1

N. J. A. Sloane, Dec 03 2008

nonn

In a posting to the Number Theory List, Oct 15 2008, Kurt Foster remarks that a positive integer M is a square iff M is a quadratic residue mod p for every prime p which does not divide M. He then asks how fast the present sequence grows.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

For records see A144295, A144296. See A092419 for another version.

with(numtheory); f:=proc(n) local M,i,j,k; M:=100000; for i from 2 to M do if legendre(n,ithprime(i)) = -1 then RETURN(ithprime(i)); fi; od; -1; end;

a(n)=my(k=n+(sqrtint(4*n)+1)\2); forprime(p=2,, if(!issquare(Mod(k,p)), return(p))) \\ Charles R Greathouse IV, Aug 28 2016

from math import isqrt
from sympy.ntheory import nextprime, legendre_symbol
def A144294(n):
    k, p = n+(m:=isqrt(n))+(n>=m*(m+1)+1), 2
    while (p:=nextprime(p)):
        if legendre_symbol(k,p)==-1:
            return p # Chai Wah Wu, Oct 20 2024

A306220 a(n) is the smallest prime p such that Kronecker(-n,p) = -1.

Original entry on oeis.org

3, 5, 2, 3, 2, 13, 3, 5, 7, 3, 2, 5, 2, 11, 7, 3, 5, 5, 2, 11, 2, 3, 5, 13, 3, 11, 2, 3, 2, 7, 3, 5, 5, 3, 2, 7, 2, 5, 7, 3, 13, 5, 2, 7, 2, 3, 5, 5, 3, 7, 2, 3, 2, 13, 3, 11, 5, 3, 2, 7, 2, 5, 5, 3, 7, 19, 2, 5, 2, 3, 7, 5, 3, 7, 2, 3, 2, 5, 3, 11, 7, 3, 2, 13, 2, 7, 5

Offset: 1

Jianing Song, Jan 29 2019

nonn

A companion sequence to A092419.

Conjecture: lim sup log(a(n))/log(n) = 0. For example, it seems that log(a(n))/log(n) < 0.5 for all n > 1364.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A092419.

See A306224 for another version.

# This requires Maple 2016 or later
f:= proc(n) local p;
  p:= 2;
  while NumberTheory:-KroneckerSymbol(-n,p) <> -1 do p:= nextprime(p) od:
  p
end proc:
map(f, [$1..100]); # Robert Israel, Feb 17 2019

a[n_] := For[p = 2, True, p = NextPrime[p], If[KroneckerSymbol[-n, p] == -1, Return[p]]];
Array[a, 100] (* Jean-François Alcover, Jun 18 2020 *)

a(n) = forprime(p=2, , if(kronecker(-n, p)<0, return(p)))

a(n) = 2 if and only if n == 3, 5 (mod 8). See A047621.

a(n) = 3 if and only if n == 1, 4, 7, 10, 16, 22 (mod 24).

A373088 a(n) = min{k : KroneckerSymbol(n, k) = -1} if n is not a square, 0 otherwise.

Original entry on oeis.org

0, 0, 3, 2, 0, 2, 7, 5, 3, 0, 7, 2, 5, 2, 3, 13, 0, 3, 5, 2, 3, 2, 5, 3, 7, 0, 3, 2, 5, 2, 11, 7, 3, 5, 7, 2, 0, 2, 3, 11, 7, 3, 5, 2, 3, 2, 11, 3, 5, 0, 3, 2, 5, 2, 7, 7, 3, 5, 5, 2, 13, 2, 3, 5, 0, 3, 7, 2, 3, 2, 13, 3, 5, 5, 3, 2, 7, 2, 5, 11, 3, 0, 5, 2

Offset: 0

Peter Luschny, May 26 2024

nonn

Similar: A092419, A144294.

Cf. A372728.

K := (n, k) -> NumberTheory:-KroneckerSymbol(n, k):
a := proc(n) if issqr(n) then return 0 fi;
local k; k := 0;
    while true do
        if K(n, k) = -1 then return k fi;
        k := k + 1;
od; -1; end:
seq(a(n), n = 0..83);

a(n) = if (issquare(n), 0, my(k=1); while (kronecker(n,k) != -1, k++); k); \\ Michel Marcus, May 31 2024

def A373088(n):
    if is_square(n): return 0
    k = 0
    while True:
        if kronecker_symbol(n, k) == -1:
            return k
        k += 1
    return k
print([A373088(n) for n in range(83)])

If n is not a square then a(n) is a prime number.

cp's OEIS Frontend

A067073 Records in A092419.

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A070040 Where records occur in A092419.

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A232931 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).

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A144294 Let k = n-th nonsquare = A000037(n); then a(n) = smallest prime p such that k is not a square mod p.

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A306220 a(n) is the smallest prime p such that Kronecker(-n,p) = -1.

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A373088 a(n) = min{k : KroneckerSymbol(n, k) = -1} if n is not a square, 0 otherwise.

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