A368415 Array read by ascending antidiagonals. A(n, k) = floor((n^k + 3)*(n/(2*n + 2))).

1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 3, 7, 11, 6, 1, 3, 11, 26, 31, 11, 1, 4, 16, 53, 103, 92, 22, 1, 4, 22, 93, 261, 410, 274, 43, 1, 5, 29, 151, 556, 1303, 1639, 821, 86, 1, 5, 37, 228, 1051, 3333, 6511, 6554, 2461, 171, 1, 6, 46, 329, 1821, 7354, 19996, 32553, 26215, 7382, 342, 1, 6, 56, 455, 2953

Offset

1,4

Comments

Let p be an odd prime number, then A(p, k) is the number of distinct quadratic residues mod p^k. Let m = p1^k1^*p2^k2*..*pz^kz with p1..pz odd primes, then A(p1, k1)*A(p2, k2)*..*A(pz, kz) is the number of distinct quadratic residues mod m. For 2^t*m is floor((2^t+10)*(1/6))*A(p1, k1)*A(p2, k2)*..*A(pz, kz) the number of distinct quadratic residues mod 2^t*m.

Links

Table of n, a(n) for n=1..70.

Formula

A(n, k) = n*A(n, k-1) + A(n, k-2) - n*A(n, k-3), for k > 2 and A(n, 0) = 1.

A(1, k) = 1.

A(2, k) = A005578(k).

A(3, k) = A039300(k).

A(4, k) = A363773(k).

A(5, k) = A039302(k).

A(7, k) = A039304(k).

A(8, k) = A172241(k+1)+1.

A(n, 2) = A000124(n-1), for n > 0.

Example

The array A(n, k) begins:
1,  1,   1,    1,     1,      1,       1,        1,         1,          1
1,  2,   3,    6,    11,     22,      43,       86,       171,        342
2,  4,  11,   31,    92,    274,     821,     2461,      7382,      22144
2,  7,  26,  103,   410,   1639,    6554,    26215,    104858,     419431
3, 11,  53,  261,  1303,   6511,   32553,   162761,    813803,    4069011
3, 16,  93,  556,  3333,  19996,  119973,   719836,   4319013,   25914076
4, 22, 151, 1051,  7354,  51472,  360301,  2522101,  17654704,  123582922
4, 29, 228, 1821, 14564, 116509,  932068,  7456541,  59652324,  477218589
5, 37, 323, 2953, 26573, 239149, 2152337, 19371025, 174339221, 1569052981
5, 46, 455, 4546, 45455, 454546, 4545455, 45454546, 454545455, 4545454546

Prog

(PARI) A(n, k) = (n^(k+1)+n*3)\(2*n+2)

Crossrefs

Cf. A000124, A005578, A039300, A039302, A039304, A172241, A363773.

Sequence in context: A129687 A274742 A128176 * A370075 A144963 A305632

Adjacent sequences: A368412 A368413 A368414 * A368416 A368417 A368418

Keyword

nonn,tabl

Author

Thomas Scheuerle, Dec 23 2023

Status

approved