A373751 - OEIS

%I #40 Oct 07 2024 02:04:37

%S 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,

%T 17,5,13,17,23,29,41,43,19,2,7,17,23,31,37,59,61,23,5,3,11,19,29,37,

%U 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

%N 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).

%C 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).

%H Robert G. Wilson v, <a href="/A373751/b373751.txt">Table of n, a(n) for n = 1..10011</a> (the first 141 antidiagonals, flattened).

%e Note that the cross-references are hints, not assertions about identity.

%e .

%e [ n] [ p]

%e [ 1] [ 2] [ 2,  3,  5,  7, 11, 13, 17, 19, 23, 29, ...  A000040

%e [ 2] [ 3] [ 3,  7, 13, 19, 31, 37, 43, 61, 67, 73, ...  A007645

%e [ 3] [ 5] [ 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, ...  A038872

%e [ 4] [ 7] [ 2,  7, 11, 23, 29, 37, 43, 53, 67, 71, ...  A045373

%e [ 5] [11] [ 3,  5, 11, 23, 31, 37, 47, 53, 59, 67, ...  A056874

%e [ 6] [13] [ 3, 13, 17, 23, 29, 43, 53, 61, 79, 101, ..  A038883

%e [ 7] [17] [ 2, 13, 17, 19, 43, 47, 53, 59, 67, 83, ...  A038889

%e [ 8] [19] [ 5,  7, 11, 17, 19, 23, 43, 47, 61, 73, ...  A106863

%e [ 9] [23] [ 2,  3, 13, 23, 29, 31, 41, 47, 59, 71, ...  A296932

%e [10] [29] [ 5,  7, 13, 23, 29, 53, 59, 67, 71, 83, ...  A038901

%e [11] [31] [ 2,  5,  7, 19, 31, 41, 47, 59, 67, 71, ...  A267481

%e [12] [37] [ 3,  7, 11, 37, 41, 47, 53, 67, 71, 73, ...  A038913

%e [13] [41] [ 2,  5, 23, 31, 37, 41, 43, 59, 61, 73, ...  A038919

%e [14] [43] [11, 13, 17, 23, 31, 41, 43, 47, 53, 59, ...  A106891

%e [15] [47] [ 2,  3,  7, 17, 37, 47, 53, 59, 61, 71, ...  A267601

%e [16] [53] [ 7, 11, 13, 17, 29, 37, 43, 47, 53, 59, ...  A038901

%e [17] [59] [ 3,  5,  7, 17, 19, 29, 41, 53, 59, 71, ...  A374156

%e [18] [61] [ 3,  5, 13, 19, 41, 47, 61, 73, 83, 97, ...  A038941

%e [19] [67] [17, 19, 23, 29, 37, 47, 59, 67, 71, 73, ...  A106933

%e [20] [71] [ 2,  3,  5, 19, 29, 37, 43, 71, 73, 79, ...

%e [21] [73] [ 2,  3, 19, 23, 37, 41, 61, 67, 71, 73, ...  A038957

%e [22] [79] [ 2,  5, 11, 13, 19, 23, 31, 67, 73, 79, ...

%e [23] [83] [ 3,  7, 11, 17, 23, 29, 31, 37, 41, 59, ...

%e [24] [89] [ 2,  5, 11, 17, 47, 53, 67, 71, 73, 79, ...  A038977

%e [25] [97] [ 2,  3, 11, 31, 43, 47, 53, 61, 73, 79, ...  A038987

%e .

%e Prime(n) is a term of row n because for all n >= 1, n is a quadratic residue mod n.

%p 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);

%t 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}]

%t (* 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}] *)

%o (SageMath)  # The function 'is_quadratic_residue' is defined in A373748.

%o def A373751_row(n, len):

%o     return [a for a in range(len) if is_quadratic_residue(a, n) and is_prime(a)]

%o for p in prime_range(99): print([p], A373751_row(p, 100))

%o (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

%Y 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).

%Y Cf. A374155 (column 1), A373748.

%Y Cf. A000040, A007645, A038872, A045373, A056874, A038883, A038889, A106863, A296932, A038901, A267481, A038913, A038919, A106891, A267601, A038901, A374156, A038941, A106933, A038957, A038977, A038987.

%K nonn,tabl

%O 1,1

%A _Peter Luschny_, Jun 28 2024