A082771 Triangular array, read by rows: t(n,k) = Sum(d^k: d|n), 0<=k<n.

1, 2, 3, 2, 4, 10, 3, 7, 21, 73, 2, 6, 26, 126, 626, 4, 12, 50, 252, 1394, 8052, 2, 8, 50, 344, 2402, 16808, 117650, 4, 15, 85, 585, 4369, 33825, 266305, 2113665, 3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 4, 18, 130, 1134, 10642, 103158, 1015690, 10078254, 100390882, 1001953638

Offset

1,2

Links

T. D. Noe, Rows n=1..100, flattened

Eric Weisstein's World of Mathematics, Divisor Function

Formula

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.

t(n,0) = A000005(n), t(n,n) = A023887(n).

t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.

t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.

From R. J. Mathar, Oct 29 2006: (Start)

t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);

t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);

t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);

t(11,k) = A034524(k); t(12,k) = A034660(k). (End)

Example

From R. J. Mathar:
The triangle may be extended to a rectangular array (A319278):
1  1   1    1     1 1 1 1 1 1 1 ...
2  3   5    9    17 33 65 129 257 513 1025 ...
2  4  10   28    82 244 730 2188 6562 19684 59050 ...
3  7  21   73   273 1057 4161 16513 65793 262657 1049601 ...
2  6  26  126   626 3126 15626 78126 390626 1953126 9765626 ...
4 12  50  252  1394 8052 47450 282252 1686434 10097892 60526250 ...
2  8  50  344  2402 16808 117650 823544 5764802 40353608 282475250 ...
4 15  85  585  4369 33825 266305 2113665 16843009 134480385 1074791425 ...
3 13  91  757  6643 59293 532171 4785157 43053283 387440173 3486843451 ...
4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638...

Maple

T:= (n, k)-> numtheory[sigma][k](n):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Oct 25 2024

Mathematica

T[n_, k_] := DivisorSigma[k, n];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

Crossrefs

Cf. A000005, A000203, A001157, A001158, A001159, A001160.

Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963.

Cf. A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.

Sequence in context: A207997 A304489 A034800 * A127157 A236406 A247497

Adjacent sequences: A082768 A082769 A082770 * A082772 A082773 A082774

Keyword

nonn,tabl,easy

Author

Reinhard Zumkeller, May 21 2003

Extensions

Corrected by R. J. Mathar, Dec 05 2006

Status

approved