This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A082771 #35 May 09 2025 07:14:06
%S A082771 1,2,3,2,4,10,3,7,21,73,2,6,26,126,626,4,12,50,252,1394,8052,2,8,50,
%T A082771 344,2402,16808,117650,4,15,85,585,4369,33825,266305,2113665,3,13,91,
%U A082771 757,6643,59293,532171,4785157,43053283,4,18,130,1134,10642,103158,1015690,10078254,100390882,1001953638
%N A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.
%H A082771 T. D. Noe, <a href="/A082771/b082771.txt">Rows n=1..100, flattened</a>
%H A082771 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.
%F A082771 t(n, k) = Product(((p^((e(n, p)+1)*k))-1)/(p^k-1): n=Product(p^e(n, p): p prime)), 0<=k<n.
%F A082771 t(n,0) = A000005(n), t(n,n) = A023887(n).
%F A082771 t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.
%F A082771 t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.
%F A082771 From _R. J. Mathar_, Oct 29 2006: (Start)
%F A082771 t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);
%F A082771 t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);
%F A082771 t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);
%F A082771 t(11,k) = A034524(k); t(12,k) = A034660(k). (End)
%e A082771 From _R. J. Mathar_, Dec 06 2006 (Start):
%e A082771 The triangle may be extended to a rectangular array (A319278):
%e A082771 1 1 1 1 1 1 1 1 1 1 1 ...
%e A082771 2 3 5 9 17 33 65 129 257 513 1025 ...
%e A082771 2 4 10 28 82 244 730 2188 6562 19684 59050 ...
%e A082771 3 7 21 73 273 1057 4161 16513 65793 262657 1049601 ...
%e A082771 2 6 26 126 626 3126 15626 78126 390626 1953126 9765626 ...
%e A082771 4 12 50 252 1394 8052 47450 282252 1686434 10097892 60526250 ...
%e A082771 2 8 50 344 2402 16808 117650 823544 5764802 40353608 282475250 ...
%e A082771 4 15 85 585 4369 33825 266305 2113665 16843009 134480385 1074791425 ...
%e A082771 3 13 91 757 6643 59293 532171 4785157 43053283 387440173 3486843451 ...
%e A082771 4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638... (End)
%p A082771 T:= (n,k)-> numtheory[sigma][k](n):
%p A082771 seq(seq(T(n,k), k=0..n-1), n=1..10); # _Alois P. Heinz_, Oct 25 2024
%t A082771 T[n_, k_] := DivisorSigma[k, n];
%t A082771 Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Dec 16 2021 *)
%o A082771 (PARI) row(n) = {my(f = factor(n)); vector(n, k, sigma(f, k-1));} \\ _Amiram Eldar_, May 09 2025
%Y A082771 Cf. A000005, A000203, A001157, A001158, A001159, A001160.
%Y A082771 Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963.
%Y A082771 Cf. A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.
%K A082771 nonn,tabl,easy
%O A082771 1,2
%A A082771 _Reinhard Zumkeller_, May 21 2003
%E A082771 Corrected by _R. J. Mathar_, Dec 05 2006