A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717

Offset: 0

Author

Omar E. Pol, Sep 09 2018

Keywords

• nonn
• tabl
• easy

Comments

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

Examples

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...
		

Crossrefs

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.

Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.

Main diagonal gives A319074.

Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.

Cf. A000040, A000203, A006093, A319075.

Programs

PARI

T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

Formula

T(n,k) = A000203(A000040(k)^n).

T(n,k) = Sum_{j=0..n} A000040(k)^j.

T(n,k) = Sum_{j=0..n} A319075(j,k).

T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).

T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).