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%I #12 Jan 07 2024 15:44:52
%S 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,
%T 243,16,125,13,7,1,0,128,729,32,625,35,49,2,1,0,256,2187,64,3125,97,
%U 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
%N Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.
%H Andrew Howroyd, <a href="/A322080/b322080.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F G.f. of column k: Sum_{j>=1} prime(j)^k*x^prime(j)/(1 - x^prime(j)).
%e Square array begins:
%e 0, 0, 0, 0, 0, 0, ...
%e 1, 2, 4, 8, 16, 32, ...
%e 1, 3, 9, 27, 81, 243, ...
%e 1, 2, 4, 8, 16, 32, ...
%e 1, 5, 25, 125, 625, 3125, ...
%e 2, 5, 13, 35, 97, 275, ...
%t Table[Function[k, Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%t 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
%o (PARI) T(n,k)={vecsum([p^k | p<-factor(n)[,1]])}
%o for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 26 2018
%Y Columns k=0..4 give A001221, A008472, A005063, A005064, A005065.
%Y Cf. A109974, A200768 (diagonal), A285425, A286880, A321258.
%K nonn,tabl
%O 1,5
%A _Ilya Gutkovskiy_, Nov 26 2018
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