A144294 -id:A144294 - cp's OEIS frontend

A144295 Where records occur in A144294.

1, 2, 4, 12, 82, 231, 380, 2990, 20954, 40953, 42852, 182492, 859647, 8039644, 11990003, 78269808, 3263358159, 6941215856

Offset: 1

N. J. A. Sloane, Dec 03 2008

Cf. A144294, A144296.

f(n)=my(k=n+(sqrtint(4*n)+1)\2); forprime(p=2, , if(!issquare(Mod(k, p)), return(p))); \\ A144294
lista(nn) = {my(v=vector(nn, n, f(n))); my(m=0, nm=0); for (n=1, nn, nm = v[n]; if (nm > m, print1(n, ", "); m = nm;););} \\ Michel Marcus, Jun 25 2021

a(9)-a(12) from R. J. Mathar, Dec 04 2008
a(13)-a(15) from Michel Marcus, Jun 25 2021
a(16)-a(18) from Michael S. Branicky, Sep 30 2024

A144296 Records in A144294.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 31, 41, 43, 47, 53, 71, 79, 89, 103, 127, 131, 137

Offset: 1

N. J. A. Sloane, Dec 03 2008

Cf. A144294, A144295.

f(n)=my(k=n+(sqrtint(4*n)+1)\2); forprime(p=2, , if(!issquare(Mod(k, p)), return(p))); \\ A144294
lista(nn) = {my(v=vector(nn, n, f(n))); my(m=0, nm=0); for (n=1, nn, nm = v[n]; if (nm > m, print1(nm, ", "); m = nm;););} \\ Michel Marcus, Jun 25 2021

a(9)-a(12) from R. J. Mathar, Dec 04 2008
a(13)-a(15) from Michel Marcus, Jun 25 2021
a(16)-a(18) from Michael S. Branicky, Sep 30 2024

A092419 Let k = n-th nonsquare = A000037(n); then a(n) = smallest prime p such that the Kronecker-Jacobi symbol K(k,p) = -1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 16 2008

nonn

Maple calls K(k,p) the Legendre symbol.

The old entry with this sequence number was a duplicate of A024356.

H. Cohen, A Course in Computational Number Theory, Springer, 1996 (p. 28 defines the Kronecker-Jacobi symbol).

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

Cf. A000037. Records: A067073, A070040. See A144294 for another version.

with(numtheory); f:=proc(n) local M,i,j,k; M:=100000; for i from 1 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(kronecker(k,p)<0, return(p))) \\ Charles R Greathouse IV, Aug 28 2016

Definition corrected Dec 03 2008

A094928 Let p = n-th prime == 1 mod 8 (A007519); a(n) = smallest prime q such that p is not a square mod q.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 19 2004

			n=3, p = 73, a(3) = q = 5: Legendre(73,5) = -1.

M. Kneser, Quadratische Formen, Springer, 2002; see Hilfssatz 18.3.

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

Cf. A007519, A002224, A144294, A269704.

Subsequence of A094929.

f:= proc(p) local q;
     q:= 3:
     do
      if numtheory:-quadres(p,q) = -1 then return q fi;
      q:= nextprime(q);
     od;
end proc:
map(f, select(isprime, [seq(p,p=1..10000,8)])); # Robert Israel, May 06 2019

f[n_] := Prime[ Position[ JacobiSymbol[n, Select[Range[3, n - 1], PrimeQ[ # ] &]], -1][[1, 1]] + 1]; f /@ Select[ Prime[ Range[435]], Mod[ #, 8] == 1 &] (* Robert G. Wilson v, Jun 23 2004 *)

a(n) = A094929(A269704(n)). - Robert Israel, May 06 2019

More terms from Robert G. Wilson v, Jun 23 2004

A377212 a(n) is the least number k that is not a quadratic residue modulo prime(n) but is a quadratic residue modulo all previous primes.

Original entry on oeis.org

2, 3, 6, 21, 15, 91, 246, 429, 1005, 399, 3094, 3045, 21099, 41155, 43059, 404754, 214230, 569130, 182919, 2190279, 860574, 9361374, 8042479, 33440551, 36915670, 11993466, 287638530, 182528031, 697126530, 78278655, 3263415285, 6941299170, 25856763139, 32968406926, 13803374706

Offset: 2

Robert Israel, Oct 19 2024

nonn

a(n) = A000037(j) for the least j such that A144294(j) = prime(n).

Such numbers k exist for all n >= 2: for example, if x is a quadratic nonresidue modulo prime(n), by the Chinese Remainder Theorem there exists k such that k == x (mod prime(n)) and k == 1 (mod prime(j)) for 1 <= j < n.

			a(4) = 6 because 6 is not a quadratic residue modulo 7, but is a quadratic residue modulo 2, 3, and 5, and no smaller number works.

Cf. A000037, A096636, A144294, A144295.

f:= proc(n) local k,p;
  if issqr(n) then return -1 fi;
  p:= 1;
  for k from 1 do
      p:= nextprime(p);
      if numtheory:-quadres(n,p) = -1 then return k fi
  od
end proc:
V:= Array(2..32): count:= 0:
for k from 2 while count < 31 do
  v:= f(k);
if v > 0 and v <= 32 and V[v] = 0 then
  V[v]:= k; count:= count+1
fi
od:
convert(V,list);

from itertools import count
from math import isqrt
from sympy.ntheory import prime, nextprime, legendre_symbol
def A377212(n):
    p = prime(n)
    for r in count(1):
        k, q = r+(m:=isqrt(r))+(r>=m*(m+1)+1), 2
        while (q:=nextprime(q)):
            if q>p or legendre_symbol(k,q)==-1:
                break
        if p==q:
            return k # Chai Wah Wu, Oct 20 2024

a(33)-a(36) from Chai Wah Wu, Oct 21 2024

A139401 If n is a square, a(n) is 0. Otherwise, a(n) is the smallest number k such that n is not a quadratic residue modulo k.

Original entry on oeis.org

0, 3, 4, 0, 3, 4, 4, 3, 0, 4, 3, 5, 5, 3, 4, 0, 3, 4, 4, 3, 8, 4, 3, 7, 0, 3, 4, 5, 3, 4, 4, 3, 5, 4, 3, 0, 5, 3, 4, 7, 3, 4, 4, 3, 7, 4, 3, 5, 0, 3, 4, 5, 3, 4, 4, 3, 5, 4, 3, 9, 7, 3, 4, 0, 3, 4, 4, 3, 7, 4, 3, 5, 5, 3, 4, 7, 3, 4, 4, 3, 0, 4, 3, 9, 8, 3, 4, 5, 3, 4, 4, 3, 5, 4, 3, 7, 5, 3, 4, 0, 3, 4, 4, 3, 9

Offset: 1

J. Lowell, Jun 09 2008, Jun 10 2008

nonn

I.e., if n is not a square, a(n) is the smallest number d for which a sequence that has a common difference of d contains n but that has no squares.
All nonzero values in this sequence are at least 3.
All nonzero values are prime powers, and every prime power except 2 appears in the sequence. This can be proved using the Chinese remainder theorem. - Franklin T. Adams-Watters, Jun 10 2011
Records of nonzero values in this sequence are in A066730.

			a(2) = 3 because there are no squares in the sequence 2, 5, 8, 11, 14, 17, 20, ...

Dan Uznanski, Table of n, a(n) for n = 1..10000

Cf. A066730, A100867, A144294, A354597.

a(n) = if (issquare(n), 0, my(k=2); while (issquare(Mod(n, k)), k++); k); \\ Michel Marcus, Jun 25 2021

import math
def A139401(n):
    if int(math.sqrt(n)) == math.sqrt(n):
        return 0
    for pp in range(2, n + 2):  # only really need to check prime powers
        residues = frozenset(pow(k, 2, pp) for k in range(pp))
        if n % pp not in residues:
            return pp  # Dan Uznanski, Jun 22 2021

More terms from John W. Layman, Jun 17 2008

New name from Franklin T. Adams-Watters, Jun 10 2011

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.

A306224 a(n) is the smallest prime p such that -n is not a square mod p.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Jan 30 2019

nonn

a(n) is the smallest odd prime p such that Kronecker(-n,p) = -1.
A companion sequence to A144294.
Conjecture: lim sup log(a(n))/log(n) = 0.

Cf. A144294.

See A306220 for another version.

a(n) = forprime(p=3, , if(kronecker(-n, p)<0, return(p)))

a(n) = 3 if and only if n == 1 (mod 3).

a(n) = 5 if and only if n == 2, 3, 8, 12 (mod 15).

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A144295 Where records occur in A144294.

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A144296 Records in A144294.

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A092419 Let k = n-th nonsquare = A000037(n); then a(n) = smallest prime p such that the Kronecker-Jacobi symbol K(k,p) = -1.

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A094928 Let p = n-th prime == 1 mod 8 (A007519); a(n) = smallest prime q such that p is not a square mod q.

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A377212 a(n) is the least number k that is not a quadratic residue modulo prime(n) but is a quadratic residue modulo all previous primes.

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A139401 If n is a square, a(n) is 0. Otherwise, a(n) is the smallest number k such that n is not a quadratic residue modulo k.

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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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A306224 a(n) is the smallest prime p such that -n is not a square mod p.

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