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
Examples
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, ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
- Index entries for sequences related to sums of divisors
Crossrefs
Programs
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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 -
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
Formula
G.f. of column k: Sum_{j>=1} prime(j)^k*x^prime(j)/(1 - x^prime(j)).