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).
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
Examples
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.
Links
- Robert G. Wilson v, Table of n, a(n) for n = 1..10011 (the first 141 antidiagonals, flattened).
Crossrefs
Programs
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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);
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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}] *) -
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
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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))
Comments