A267481 -id:A267481 - cp's OEIS frontend

A191040 Primes p that have Kronecker symbol (p|62) = 1.

3, 7, 11, 13, 29, 37, 41, 43, 47, 53, 61, 71, 83, 97, 103, 113, 139, 179, 181, 191, 193, 197, 229, 233, 251, 257, 269, 277, 281, 311, 331, 347, 359, 389, 431, 439, 461, 479, 491, 499, 503, 509, 521, 523, 557, 571, 577, 587, 593, 599, 607, 613, 617, 619, 643

Offset: 1

T. D. Noe, May 25 2011

Originally incorrectly named "primes which are squares (mod 62)", which is sequence A267481. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023, A191025, A191026, A191028, A191029, A191032, A191034, A191036, A191037.

[p: p in PrimesUpTo(643) | KroneckerSymbol(p, 62) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,62]==1&]

select(p->kronecker(p, 62)==1&&isprime(p), [1..1000]) \\ This is to provide a generic characteristic function ("is_A191040") as 1st arg of select(), there are other ways to produce the sequence more efficiently. - M. F. Hasler, Jan 15 2016

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

Peter Luschny, Jun 28 2024

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).

			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.

Robert G. Wilson v, Table of n, a(n) for n = 1..10011 (the first 141 antidiagonals, flattened).

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.

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);

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}] *)

A373751_row(n, LIM=99)={ my(q=prime(n)); [p | p <- primes([1,LIM]), issquare( Mod(p, q))] } \\ M. F. Hasler, Jun 29 2024

# 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))

cp's OEIS Frontend

A191040 Primes p that have Kronecker symbol (p|62) = 1.

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