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

0,2

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.

Links

Table of n, a(n) for n=0..59.

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).

Example

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, ...
...

Prog

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

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.

Sequence in context: A110814 A275599 A210038 * A286513 A193970 A274510

Adjacent sequences: A319073 A319074 A319075 * A319077 A319078 A319079

Keyword

nonn,tabl,easy

Author

Omar E. Pol, Sep 09 2018

Status

approved