

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
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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



EXAMPLE

Note that the crossreferences 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.
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

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


KEYWORD



AUTHOR



STATUS

approved



