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%I #30 Sep 18 2018 09:17:43
%S 1,3,1,7,4,1,15,13,6,1,31,40,31,8,1,63,121,156,57,12,1,127,364,781,
%T 400,133,14,1,255,1093,3906,2801,1464,183,18,1,511,3280,19531,19608,
%U 16105,2380,307,20,1,1023,9841,97656,137257,177156,30941,5220,381,24,1,2047,29524,488281,960800,1948717
%N 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.
%C 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.
%F T(n,k) = A000203(A000040(k)^n).
%F T(n,k) = Sum_{j=0..n} A000040(k)^j.
%F T(n,k) = Sum_{j=0..n} A319075(j,k).
%F T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
%F T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).
%e The corner of the square array is as follows:
%e A126646 A003462 A003463 A023000 A016123 A091030 A091045
%e A000012 1, 1, 1, 1, 1, 1, 1, ...
%e A008864 3, 4, 6, 8, 12, 14, 18, ...
%e A060800 7, 13, 31, 57, 133, 183, 307, ...
%e A131991 15, 40, 156, 400, 1464, 2380, 5220, ...
%e A131992 31, 121, 781, 2801, 16105, 30941, 88741, ...
%e A131993 63, 364, 3906, 19608, 177156, 402234, 1508598, ...
%e ....... 127, 1093, 19531, 137257, 1948717, 5229043, 25646167, ...
%e ....... 255, 3280, 97656, 960800, 21435888, 67977560, 435984840, ...
%e ....... 511, 9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
%e ...
%o (PARI) T(n, k) = sigma(prime(k)^n); \\ _Michel Marcus_, Sep 13 2018
%Y Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
%Y Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
%Y Main diagonal gives A319074.
%Y Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
%Y Cf. A000040, A000203, A006093, A319075.
%K nonn,tabl,easy
%O 0,2
%A _Omar E. Pol_, Sep 09 2018