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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.
2
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.
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:
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
KEYWORD
nonn,tabl,easy
AUTHOR
Omar E. Pol, Sep 09 2018
STATUS
approved