A369638 -id:A369638 - cp's OEIS frontend

A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 1, 6, 0, 0, 0, 0, 4, 0, 5, 7, 0, 0, 0, 0, 4, 0, 1, 1, 8, 0, 0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 0, 0, 4, 0, 0, 0, 16, 6, 10, 0, 0, 0, 0, 4, 0, 0, 0, 32, 5, 7, 11, 0, 0, 0, 0, 4, 0, 0, 0, 80, 1, 1, 1, 12

Offset: 0

Alois P. Heinz, Jun 06 2015

			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,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  4,  4,  4,  4,  4,   4,   4,   4,    4,    4, ...
  5,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  6,  5,  1,  0,  0,   0,   0,   0,    0,    0, ...
  7,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ...
  9,  6,  5,  1,  0,   0,   0,   0,    0,    0, ...

Alois P. Heinz, Antidiagonals n = 0..115, flattened
J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8
Wikipedia, Arithmetic derivative

Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.
Rows n=0,1,4,8 give: A000004, A000007, A010709, A129150.
Row 15: A090636, Row 28: A090637, Row 63: A090635, Row 81: A129151, Row 128: A369638, Row 1024: A214800, Row 15625: A129152.
Main diagonal gives A185232.
Antidiagonal sums give A258652.
Cf. also A328383.

d:= n-> n*add(i[2]/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);

d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* Jean-François Alcover, Apr 27 2017, translated from Maple *)

A(n,k) = A003415^k(n).

A369652 Maximal digit in the primorial base representation of the n-th arithmetic derivative of 128.

Original entry on oeis.org

4, 4, 6, 2, 6, 11, 8, 7, 11, 11, 8, 10, 15, 15, 12, 18, 17, 30, 28, 22, 21, 37, 28, 38, 42, 33, 22, 52, 43, 56, 47, 49, 57, 60, 55, 54, 58, 70, 52, 66, 68, 57, 63, 86, 58, 88, 92, 66, 78, 95, 85, 52, 102, 70, 111, 57, 117, 99, 136, 104, 129, 110, 146, 127, 135, 132, 131, 129, 126, 145, 112, 150, 128, 129, 154, 161, 145

Offset: 0

Antti Karttunen, Feb 04 2024

nonn

This sequence relates to the question whether A327969(128) has a positive integer value, or whether it is -1 by the escape clause.

Note that when iterating the map k -> k' from A276086(A369638(4)) = A276086(15232) = 3299611946113357875 onward, the maximal exponent in the prime factorization (A051903) keeps on decreasing until it reaches 1 at the fifth iteration, and then stays as 1 for three more iterations (with k then 38863666759992439 = 643*60441161368573), but then, alas, on the next iteration, k' = 60441161369216 = 2^7 * 472196573197.

			The third arithmetic derivative (A099306) of 128 is 5056, which in primorial base (A049345) is written as 220220, therefore a(3) = 2.
The fourth arithmetic derivative (A258644) of 128 is 15232, which in primorial base is written as 663320, therefore a(4) = 6.

Sean A. Irvine, Table of n, a(n) for n = 0..142
Index entries for sequences related to primorial base

Cf. A003415, A049345, A099306, A258644, A328114, A369638.

Cf. also A327969.

a(n) = A328114(A369638(n)).

cp's OEIS Frontend

A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula