cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A373751 Array read by ascending antidiagonals: p is a term of row A(n) if and only if p is a prime and p is a quadratic residue modulo prime(n).

Original entry on oeis.org

2, 3, 3, 5, 7, 5, 2, 11, 13, 7, 3, 7, 19, 19, 11, 3, 5, 11, 29, 31, 13, 2, 13, 11, 23, 31, 37, 17, 5, 13, 17, 23, 29, 41, 43, 19, 2, 7, 17, 23, 31, 37, 59, 61, 23, 5, 3, 11, 19, 29, 37, 43, 61, 67, 29, 2, 7, 13, 17, 43, 43, 47, 53, 71, 73, 31, 3, 5, 13, 23, 19, 47, 53, 53, 67, 79, 79, 37
Offset: 1

Views

Author

Peter Luschny, Jun 28 2024

Keywords

Comments

p is a term of A(n) <=> p is prime and there exists an integer q such that q^2 is congruent to p modulo prime(n).

Examples

			Note that the cross-references are hints, not assertions about identity.
.
[ n] [ p]
[ 1] [ 2] [ 2,  3,  5,  7, 11, 13, 17, 19, 23, 29, ...  A000040
[ 2] [ 3] [ 3,  7, 13, 19, 31, 37, 43, 61, 67, 73, ...  A007645
[ 3] [ 5] [ 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, ...  A038872
[ 4] [ 7] [ 2,  7, 11, 23, 29, 37, 43, 53, 67, 71, ...  A045373
[ 5] [11] [ 3,  5, 11, 23, 31, 37, 47, 53, 59, 67, ...  A056874
[ 6] [13] [ 3, 13, 17, 23, 29, 43, 53, 61, 79, 101, ..  A038883
[ 7] [17] [ 2, 13, 17, 19, 43, 47, 53, 59, 67, 83, ...  A038889
[ 8] [19] [ 5,  7, 11, 17, 19, 23, 43, 47, 61, 73, ...  A106863
[ 9] [23] [ 2,  3, 13, 23, 29, 31, 41, 47, 59, 71, ...  A296932
[10] [29] [ 5,  7, 13, 23, 29, 53, 59, 67, 71, 83, ...  A038901
[11] [31] [ 2,  5,  7, 19, 31, 41, 47, 59, 67, 71, ...  A267481
[12] [37] [ 3,  7, 11, 37, 41, 47, 53, 67, 71, 73, ...  A038913
[13] [41] [ 2,  5, 23, 31, 37, 41, 43, 59, 61, 73, ...  A038919
[14] [43] [11, 13, 17, 23, 31, 41, 43, 47, 53, 59, ...  A106891
[15] [47] [ 2,  3,  7, 17, 37, 47, 53, 59, 61, 71, ...  A267601
[16] [53] [ 7, 11, 13, 17, 29, 37, 43, 47, 53, 59, ...  A038901
[17] [59] [ 3,  5,  7, 17, 19, 29, 41, 53, 59, 71, ...  A374156
[18] [61] [ 3,  5, 13, 19, 41, 47, 61, 73, 83, 97, ...  A038941
[19] [67] [17, 19, 23, 29, 37, 47, 59, 67, 71, 73, ...  A106933
[20] [71] [ 2,  3,  5, 19, 29, 37, 43, 71, 73, 79, ...
[21] [73] [ 2,  3, 19, 23, 37, 41, 61, 67, 71, 73, ...  A038957
[22] [79] [ 2,  5, 11, 13, 19, 23, 31, 67, 73, 79, ...
[23] [83] [ 3,  7, 11, 17, 23, 29, 31, 37, 41, 59, ...
[24] [89] [ 2,  5, 11, 17, 47, 53, 67, 71, 73, 79, ...  A038977
[25] [97] [ 2,  3, 11, 31, 43, 47, 53, 61, 73, 79, ...  A038987
.
Prime(n) is a term of row n because for all n >= 1, n is a quadratic residue mod n.

Links

Crossrefs

Family: A217831 (Euclid's triangle), A372726 (Legendre's triangle), A372877 (Jacobi's triangle), A372728 (Kronecker's triangle), A373223 (Gauss' triangle), A373748 (quadratic residue/nonresidue modulo n).
Cf. A374155 (column 1), A373748.
Cf. A000040, A007645, A038872, A045373, A056874, A038883, A038889, A106863, A296932, A038901, A267481, A038913, A038919, A106891, A267601, A038901, A374156, A038941, A106933, A038957, A038977, A038987.

Programs

  • Maple
    A := proc(n, len) local c, L, a; a := 2; c := 0; L := NULL; while c < len do if NumberTheory:-QuadraticResidue(a, n) = 1 and isprime(a) then L := L,a; c := c + 1 fi; a := a + 1 od; [L] end: seq(print(A(ithprime(n), 10)), n = 1..25);
  • Mathematica
    f[m_, n_] := Block[{p = Prime@ m}, Union[ Join[{p}, Select[ Prime@ Range@ 22, JacobiSymbol[#, If[m > 1, p, 1]] == 1 &]]]][[n]]; Table[f[n, m -n +1], {m, 12}, {n, m, 1, -1}]
(* To read the array by descending antidiagonals, just exchange the first argument with the second in the function "f" called by the "Table"; i.e., Table[ f[m -n +1, n], {m, 12}, {n, m, 1, -1}] *)
  • PARI
    A373751_row(n, LIM=99)={ my(q=prime(n)); [p | p <- primes([1,LIM]), issquare( Mod(p, q))] } \\ M. F. Hasler, Jun 29 2024
  • SageMath
    # The function 'is_quadratic_residue' is defined in A373748.
def A373751_row(n, len):
    return [a for a in range(len) if is_quadratic_residue(a, n) and is_prime(a)]
for p in prime_range(99): print([p], A373751_row(p, 100))

A141178 Primes of the form 3*x^2+x*y-3*y^2 (as well as of the form 3*x^2+7*x*y+y^2).

Original entry on oeis.org

3, 7, 11, 37, 41, 47, 53, 67, 71, 73, 83, 101, 107, 127, 137, 139, 149, 151, 157, 173, 181, 197, 211, 223, 229, 233, 263, 269, 271, 293, 307, 317, 337, 349, 359, 367, 373, 379, 397, 419, 433, 443, 491, 509, 521, 571, 593, 599, 601, 613, 617, 619, 641, 659, 673
Offset: 1

Views

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008

Keywords

Comments

Discriminant = 37. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1

Examples

			a(3) = 11 because we can write 11 = 3*2^2+2*1-3*1^2 (or 11 = 3*1^2+7*1*1+1^2).

References

  • Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

Links

Crossrefs

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Primes in A035267.
A subsequence of (and may possibly coincide with) A038913. - R. J. Mathar, Jul 22 2008
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

  • Mathematica
    Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == 3*x^2 + x*y - 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 37 || MatchQ[Mod[#, 37], Alternatives[1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36]]&](* Jean-François Alcover, Oct 25 2016, updated Oct 30 2016 *)

A191046 Primes p that have Kronecker symbol (p|74) = 1.

Original entry on oeis.org

5, 7, 13, 19, 29, 41, 43, 47, 59, 61, 71, 73, 109, 127, 131, 137, 151, 163, 179, 223, 227, 233, 251, 263, 271, 277, 283, 331, 337, 347, 359, 367, 389, 421, 433, 461, 467, 499, 521, 523, 541, 547, 557, 563, 587, 593, 599, 601, 617, 641, 643, 653, 661, 673
Offset: 1

Views

Author

T. D. Noe, May 25 2011

Keywords

Comments

Originally incorrectly named "primes which are squares mod 74", which is sequence A038913. - M. F. Hasler, Jan 15 2016

Links

Crossrefs

Cf. A191017, A191018, A191020, A191023, A191025, A191026, A191028, A191029, A191032, A191034, A191036, A191037, A191040, A191042, A191043, A191047, A191049.

Programs

  • Magma
    [p: p in PrimesUpTo(673) | KroneckerSymbol(p, 74) eq 1]; // Vincenzo Librandi, Sep 11 2012
  • Mathematica
    Select[Prime[Range[200]], JacobiSymbol[#,74]==1&]
  • PARI
    select(p->kronecker(p, 74)==1&&isprime(p), [1..1000]) \\ This is to provide a generic characteristic function ("is_A191046") as 1st arg of select(), there are other ways to produce the sequence more efficiently. - M. F. Hasler, Jan 15 2016

Extensions

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191027 Primes that are nonzero squares mod 37.

Original entry on oeis.org

3, 7, 11, 41, 47, 53, 67, 71, 73, 83, 101, 107, 127, 137, 139, 149, 151, 157, 173, 181, 197, 211, 223, 229, 233, 263, 269, 271, 293, 307, 317, 337, 349, 359, 367, 373, 379, 397, 419, 433, 443, 491, 509, 521, 571, 593, 599, 601, 613, 617, 619, 641, 659, 673
Offset: 1

Views

Author

T. D. Noe, May 24 2011

Keywords

Comments

Primes p such that the Legendre symbol (p/37) = +1. - N. J. A. Sloane, May 25 2013

Links

Crossrefs

Except for excluding 37, identical to A038913.

Programs

  • Magma
    [p: p in PrimesUpTo(643) | JacobiSymbol(p, 37) eq 1]; // Vincenzo Librandi, Sep 10 2012
  • Mathematica
    Select[Prime[Range[200]], JacobiSymbol[#,37]==1&]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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