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 A373751 Array read by ascending antidiagonals: p is term of row A(n) if and only if p is a prime and p is a quadratic residue modulo prime(n). 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS p is term of A(n) <=> p is prime and there exists an integer q such that q^2 is congruent to p modulo prime(n). LINKS Robert G. Wilson v, Table of n, a(n) for n = 1..10011 (the first 141 antidiagonals, flattened). EXAMPLE 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 term of row n because for all n >= 1, n is a quadratic residue mod n. 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}] *) PROG (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)) (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 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. Sequence in context: A209702 A077558 A122444 * A066072 A239586 A180611 Adjacent sequences: A373748 A373749 A373750 * A373752 A373753 A373754 KEYWORD nonn,tabl AUTHOR Peter Luschny, Jun 28 2024 STATUS approved

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Last modified August 13 20:30 EDT 2024. Contains 375144 sequences. (Running on oeis4.)