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.

A191037 Primes p that have Jacobi symbol (p|58) = 1.

Original entry on oeis.org

3, 7, 11, 19, 23, 37, 43, 61, 71, 101, 103, 131, 151, 157, 163, 167, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 281, 293, 307, 313, 317, 331, 353, 379, 383, 389, 401, 421, 431, 439, 443, 457, 461, 463, 467, 487, 491, 521, 541, 563, 593, 619, 631, 647
Offset: 1

Views

Author

T. D. Noe, May 25 2011

Keywords

Comments

Originally incorrectly named "Primes which are squares mod 58", which is sequence A038901. - M. F. Hasler, Jan 15 2016

Links

Crossrefs

Cf. A191017, A191018, A191020, A191023, A191025, A191026, A191028, A191029, A191032, A191034, A191036.

Programs

  • Magma
    [p: p in PrimesUpTo(647) | KroneckerSymbol(p, 58) eq 1]; // Vincenzo Librandi, Sep 11 2012
  • Maple
    select(t -> isprime(t) and numtheory:-jacobi(t,58)=1, [seq(i,i=3..1000,2)]); # Robert Israel, Jan 15 2016
  • Mathematica
    Select[Prime[Range[200]], JacobiSymbol[#,58]==1&]
  • PARI
    select(p->kronecker(p,58)==1&&isprime(p),[1..1000]) \\ This is to provide a generic characteristic function ("is_A191037") 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

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))

A035264 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 29.

Original entry on oeis.org

1, 4, 5, 7, 9, 13, 16, 20, 23, 25, 28, 29, 35, 36, 45, 49, 52, 53, 59, 63, 64, 65, 67, 71, 80, 81, 83, 91, 92, 100, 103, 107, 109, 112, 115, 116, 117, 121, 125, 139, 140, 144, 145, 149, 151, 161, 167, 169, 173, 175, 179, 180, 181, 196, 197, 199, 203, 207, 208
Offset: 1

Views

Author

N. J. A. Sloane, Dec 11 1999

Keywords

Comments

Terms seem to be exactly the numbers represented by the indefinite binary quadratic form (1, 7, 5) with discriminant 29 (Lagrange-Gauss reduced (1, 5, -1)). - Peter Luschny, Jun 24 2014

Links

Crossrefs

Cf. A038901.

Programs

  • PARI
    m=29; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

Extensions

Name corrected by Andrey Zabolotskiy, Jul 30 2020

A363482 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(-5))))).

Original entry on oeis.org

13, 23, 7, 49, 13, 83, 103, 5, 149, 1, 29, 233, 53, 23, 67, 373, 59, 1, 499, 109, 593, 643, 139, 107, 1, 863, 71, 197, 1049, 223, 1, 179, 53, 1399, 59, 1553, 71, 1, 257, 1, 1973, 2063, 431, 173, 67, 349, 2543, 1, 2749, 571, 2963, 439, 1, 3299, 683, 3533, 281, 151, 557, 1, 4153
Offset: 3

Views

Author

Mohammed Bouras, Jun 04 2023

Keywords

Comments

Conjecture: Except for 49, every term of this sequence is either a prime or 1.
The conjecture holds through n=10000. The set of ordered primes appear to match A038901. - Bill McEachen, Jul 08 2025

Examples

			For n=3, 1/(2 - 3/(-5)) = 5/13, so a(3) = 13.
For n=4, 1/(2 - 3/(3 - 4/(-5))) = 19/23, so a(4) = 23.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(-5)))) = 11/7, so a(5) = 7.

Links

Crossrefs

Cf. A006530, A038901, A051403.
Cf. A362086, A363102.

Programs

  • PARI
    lf(n) = sum(k=0, n-1, k!); \\ A003422
f(n) = (n+2)*lf(n+1)/2; \\ A051403
a(n) = (n^2 + 3*n - 5)/gcd(n^2 + 3*n - 5, 5*f(n-3) + n*f(n-4)); \\ Michel Marcus, Jun 06 2023

Formula

a(n) = (n^2 + 3*n - 5)/gcd(n^2 + 3*n - 5, 5*A051403(n-3) + n*A051403(n-4)).
Except for n=6, if gpf(n^2 + 3*n - 5) > n, then we have:
a(n) = gpf(n^2 + 3*n - 5), where gpf = "greatest prime factor".
If a(n) = a(m) and n < m < a(n), then we have:
a(n) = n + m + 3.
a(n) divides gcd(n^2 + 3*n - 5, m^2 + 3*m - 5).
Showing 1-4 of 4 results.

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

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