A034794 -id:A034794 - cp's OEIS frontend

A092581 a(n) is the least prime such that a(n-1) is a quadratic non-residue of a(n).

2, 3, 5, 7, 11, 13, 19, 23, 31, 37, 43, 47, 59, 61, 67, 71, 79, 83, 89, 101, 103, 107, 109, 127, 131, 137, 149, 151, 157, 163, 167, 179, 181, 191, 199, 227, 229, 239, 251, 257, 263, 271, 277, 283, 307, 311, 331, 347, 349, 359, 367, 373, 379, 383, 409, 431, 439

Offset: 1

Gary W. Adamson, Feb 29 2004

nonn

Paulo Ribenboim, "The Little Book of Big Primes", Springer-Verlag, 1991, p. 28.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A034794.

first Needs[ "NumberTheory`NumberTheoryFunctions`" ] then f[n_] := Block[{k = PrimePi[n] + 1}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; NestList[f, 2, 56] (* Robert G. Wilson v, Mar 16 2004 *)

"If p>2 does not divide a and if there exists an integer b such that a is congruent to b^2 (mod p), then a is called a quadratic residue modulo p; otherwise, it is a nonquadratic residue modulo p". (p. 28, Ribenboim)

More terms from Robert G. Wilson v, Mar 16 2004

a(17) corrected by T. D. Noe, Aug 28 2007

A140292 a(n) is a square mod a(n-1), a(n) > a(n-1) and a(n) semiprime.

Original entry on oeis.org

4, 9, 10, 14, 15, 21, 22, 25, 26, 35, 39, 49, 51, 55, 69, 82, 86, 87, 91, 95, 106, 115, 119, 121, 122, 123, 133, 134, 143, 146, 155, 159, 166, 169, 178, 183, 187, 202, 203, 219, 235, 249, 253, 254, 262, 265, 274, 278, 287, 289, 291, 295, 299, 302, 303, 309, 327

Offset: 1

Jonathan Vos Post, May 24 2008

Cf. A001358, A034794, A088190.

isqResid := proc(n,modp) local x ; for x from 1 to floor(modp/2) do if x^2 mod modp = n mod modp then RETURN(true) ; fi ; od: RETURN(false) ; end: isA001358 := proc(n) RETURN( numtheory[bigomega](n)= 2) ; end: A140292 := proc(n) option remember ; local a; if n = 1 then 4; else for a from A140292(n-1)+1 do if isA001358(a) and isqResid(a,A140292(n-1)) then RETURN(a) ; fi ; od ; fi ; end: seq(A140292(n),n=1..80) ; # R. J. Mathar, May 31 2008

quadResQ[n_, p_] := Module[{x}, For[x = 1, x <= Floor[p/2], x++, If[Mod[x^2, p] == Mod[n, p], Return[True]]]; Return[False]];
semiprimeQ[n_] := PrimeOmega[n] == 2;
a[n_] := a[n] = Module[{k}, If[n == 1, 4, For[k = a[n - 1] + 1, True, k++, If[semiprimeQ[k] && quadResQ[k, a[n - 1]], Return[k]]]]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 28 2024, after R. J. Mathar *)

Corrected and extended by R. J. Mathar, May 31 2008

A359633 a(n) is the least prime > a(n-1) such that a(n-1) and a(n) are quadratic residues mod each other.

Original entry on oeis.org

2, 7, 29, 53, 59, 137, 139, 173, 179, 193, 197, 223, 241, 251, 317, 353, 383, 389, 409, 419, 457, 461, 467, 541, 557, 563, 593, 601, 607, 701, 743, 761, 769, 773, 787, 797, 811, 853, 857, 859, 881, 883, 929, 937, 941, 947, 977, 991, 1009, 1013, 1019, 1033, 1039, 1049, 1051, 1097, 1129, 1153, 1171

Offset: 1

Robert Israel, Jan 07 2023

nonn

Quadratic reciprocity says that for odd primes p and q, if p is a quadratic residue mod q then q is a quadratic residue mod p except in the case where p and q are both congruent to 3 (mod 4), in which case they can't both be quadratic residues mod each other. Thus if a(n-1) == 1 (mod 4), a(n) is the least prime > a(n-1) that is a quadratic residue mod a(n-1), while if a(n-1) == 3 (mod 4), a(n) is the least prime > a(n-1) that is congruent to 1 (mod 4) and is a quadratic residue mod a(n-1).

			a(3) = 29 because a(2) = 7, 29 is a quadratic residue mod 7 and 7 is a quadratic residue mod 29, and 29 is the least prime > 7 that works.

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

Cf. A034794, A034795.

f:= proc(p) local q;
   q:= p;
   do
     q:= nextprime(q);
     if NumberTheory:-QuadraticResidue(q,p) = 1 and NumberTheory:-QuadraticResidue(p,q) = 1  then return q fi
   od
end proc:
A[1]:= 2: for i from 2 to 100 do A[i]:= f(A[i-1]) od:
seq(A[i], i=1..100);

A177051 A subsequence of the Fibonacci sequence such that a(n) is a quadratic residue mod a(n+1).

Original entry on oeis.org

1, 2, 34, 55, 89, 233, 377, 1597, 17711, 28657, 121393, 317811, 1346269, 3524578, 5702887, 24157817, 39088169, 63245986, 433494437, 2971215073, 53316291173, 591286729879, 956722026041, 2504730781961, 4052739537881, 17167680177565, 308061521170129, 5527939700884757, 61305790721611591, 99194853094755497, 1779979416004714189

Offset: 1

Michel Lagneau, Dec 09 2010

nonn

			2, 34 and 55 are in the sequence because L(2/34) = L(34/55) = 1 where L(a/b) is the Legendre symbol of a and b, which is defined to be 1 if a is a quadratic residue (mod b) and -1 if a is a quadratic non-residue (mod b).

Cf. A034794, A000045.

with(combinat, fibonacci):k:=1:pr0:=fibonacci(k):for n from k+1 to 100 do:pr:=fibonacci(n):if
  quadres(pr0,pr)=1then pr0:=pr:printf(`%d, `,pr):else fi:od:

print1(k=1); for(n=3,100,t=fibonacci(n); if(issquare(Mod(k,t)), print1(", "k=t))) \\ Charles R Greathouse IV, Jan 09 2013

Definition corrected by Michel Lagneau

cp's OEIS Frontend

A092581 a(n) is the least prime such that a(n-1) is a quadratic non-residue of a(n).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A140292 a(n) is a square mod a(n-1), a(n) > a(n-1) and a(n) semiprime.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Extensions

A359633 a(n) is the least prime > a(n-1) such that a(n-1) and a(n) are quadratic residues mod each other.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A177051 A subsequence of the Fibonacci sequence such that a(n) is a quadratic residue mod a(n+1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Extensions