A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

0, 0, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 9, 2, 1, 0, 16, 27, 4, 5, 2, 0, 32, 81, 8, 25, 5, 1, 0, 64, 243, 16, 125, 13, 7, 1, 0, 128, 729, 32, 625, 35, 49, 2, 1, 0, 256, 2187, 64, 3125, 97, 343, 4, 3, 2, 0, 512, 6561, 128, 15625, 275, 2401, 8, 9, 7, 1, 0, 1024, 19683, 256, 78125, 793, 16807, 16, 27, 29, 11, 2

Offset

1,5

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Index entries for sequences related to sums of divisors

Formula

G.f. of column k: Sum_{j>=1} prime(j)^k*x^prime(j)/(1 - x^prime(j)).

Example

Square array begins:
  0,  0,   0,    0,    0,     0,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  3,   9,   27,   81,   243,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  5,  25,  125,  625,  3125,  ...
  2,  5,  13,   35,   97,   275,  ...

Mathematica

Table[Function[k, Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[Prime[j]^k x^Prime[j]/(1 - x^Prime[j]), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

Prog

(PARI) T(n, k)={vecsum([p^k | p<-factor(n)[, 1]])}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

Crossrefs

Columns k=0..4 give A001221, A008472, A005063, A005064, A005065.

Cf. A109974, A200768 (diagonal), A285425, A286880, A321258.

Sequence in context: A255528 A201701 A131667 * A361958 A334122 A086802

Adjacent sequences: A322077 A322078 A322079 * A322081 A322082 A322083

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Nov 26 2018

Status

approved