|
| |
|
|
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
|
|
|
|
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.
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
|
| |
|
|
