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A082771
Triangular array, read by rows: t(n,k) = Sum(d^k: d|n), 0<=k<n.
7
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
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
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 *)
KEYWORD
nonn,tabl,easy,changed
AUTHOR
Reinhard Zumkeller, May 21 2003
EXTENSIONS
Corrected by R. J. Mathar, Dec 05 2006
STATUS
approved