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A258850
A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
16
0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 4, 0, 0, 0, 1, 4, 5, 0, 0, 0, 0, 4, 3, 6, 0, 0, 0, 0, 4, 2, 7, 7, 0, 0, 0, 0, 4, 1, 4, 4, 8, 0, 0, 0, 0, 4, 0, 4, 4, 12, 9, 0, 0, 0, 0, 4, 0, 4, 4, 20, 12, 10, 0, 0, 0, 0, 4, 0, 4, 4, 32, 20, 11, 11, 0, 0, 0, 0, 4, 0, 4, 4, 80, 32, 5, 5, 12
OFFSET
0,6
LINKS
FORMULA
A(n,k) = A258851^k(n).
A(A259016(n,k),k) = n.
A(A258975(n),n) = 1.
EXAMPLE
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
3, 2, 1, 0, 0, 0, 0, 0, 0, 0, ...
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
5, 3, 2, 1, 0, 0, 0, 0, 0, 0, ...
6, 7, 4, 4, 4, 4, 4, 4, 4, 4, ...
7, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
8, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
9, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
MAPLE
with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
seq(seq(A(n, h-n), n=0..h), h=0..14);
MATHEMATICA
d[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; d[0] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[Table[A[n, h-n], {n, 0, h}], {h, 0, 14}] // Flatten (* Jean-François Alcover, Apr 24 2016, adapted from Maple *)
CROSSREFS
Rows n=0,1,4,8 give: A000004, A000007, A010709, A258848.
Antidiagonal sums give A258847.
Main diagonal gives A258849.
Sequence in context: A153036 A258651 A350530 * A182114 A122950 A374766
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 12 2015
STATUS
approved