A084890 Triangular array, read by rows: T(n,k) = arithmetic derivative of n*k, 1<=k<=n.

0, 1, 4, 1, 5, 6, 4, 12, 16, 32, 1, 7, 8, 24, 10, 5, 16, 21, 44, 31, 60, 1, 9, 10, 32, 12, 41, 14, 12, 32, 44, 80, 68, 112, 92, 192, 6, 21, 27, 60, 39, 81, 51, 156, 108, 7, 24, 31, 68, 45, 92, 59, 176, 123, 140, 1, 13, 14, 48, 16, 61, 18, 140, 75, 87, 22, 16, 44, 60, 112, 92, 156, 124, 272, 216, 244, 188, 384

Offset

1,3

Links

Table of n, a(n) for n=1..78.

Eric Weisstein's World of Mathematics, Product Rule.

Formula

T(n,k) = A003415(n*k) = n*A003415(k)+k*A003415(n), 1<=k<=n.

T(n,1) = A003415(n); n>1.

T(n,2) = A068719(n).

T(n,n) = A068720(n).

Example

................. 0
.............. 1 ... 4
........... 1 ... 5 ... 6
........ 4 .. 12 .. 16 .. 32
..... 1 ... 7 ... 8 .. 24 .. 10
.. 5 .. 16 .. 21 .. 44 .. 31 .. 60
1 ... 9 .. 10 .. 32 .. 12 .. 41 .. 14.

Mathematica

ader[n_] := ader[n] =Switch[n, 0 | 1, 0, _, If[PrimeQ[n], 1, Sum[Module[{p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]]];
T[n_, k_] := ader[n k];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 21 2021 *)

Crossrefs

Cf. A084884, A084885, A084886, A084887.

Sequence in context: A333341 A344027 A379623 * A058662 A344026 A133866

Adjacent sequences: A084887 A084888 A084889 * A084891 A084892 A084893

Keyword

nonn,tabl,changed

Author

Reinhard Zumkeller, Jun 10 2003

Status

approved