A090636 -id:A090636 - 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).

A129150 The n-th arithmetic derivative of 2^3.

Original entry on oeis.org

8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, 8592, 20096, 70464, 235072, 705280, 3023616, 13223680, 55540736, 278539264, 1392697344, 9541095424, 58609614848, 410267320320, 3397142953984, 24143851798528, 176071227916288, 1232666139967488, 9523075842834432

Offset: 0

Reinhard Zumkeller, Apr 01 2007

nonn

Conjecture: a strictly increasing sequence. - J. Lowell, Sep 10 2008
The sequence is strictly increasing because (4*n)' = 4*n + 4*n'. - David Radcliffe, Aug 19 2014
8 is the smallest integer that has a nontrivial trajectory (not going to 0 nor reduced to a fixed point as 4) under A003415, but 15 = A090636(1) has 8 as second term in its trajectory. 20 is the next larger such integer with a distinct trajectory, but has two larger predecessors, cf. A090635. - M. F. Hasler, Nov 27 2019
In general, the trajectory of p^(p+1) under A003415 has a common factor p^p, and divided by p^p it gives the trajectory of p under A129283: n -> n + n'. Here we have the case p = 2 (see A129284 for a(n)/2^2), cf. A129151 and A129152 for p = 3 and 5. - M. F. Hasler, Nov 28 2019

Charles R Greathouse IV, Table of n, a(n) for n = 0..100

Cf. A129151, A129152, A068327, A099309, A051674, A100716, A129284.

Row n = 8 of A258651.

a129150 n = a129150_list !! n
a129150_list = iterate a003415 8  -- Reinhard Zumkeller, Apr 29 2012

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; s = 2^3; Join[{s}, Table[s = dn[s], {28}]] (* T. D. Noe, Mar 07 2013 *)

A129150(n,a=8)={if(n<0, vector(-n,n, if(n>1, a=A003415(a), a)), for(n=1,n, a=A003415(a)); a)}  \\ For n<0 return the vector a[0..-n-1]. - M. F. Hasler, Nov 27 2019

a(n+1) = A003415(a(n)), a(0) = 2^3 = 8.
a(n) = A090636(n+2).
A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007
a(n) = 4*A129284(n). - M. F. Hasler, Nov 27 2019

a(21)-a(27) from Paolo P. Lava, Apr 16 2012

A090635 Trajectory of 63 under the map k -> A003415(k) (taking the arithmetic derivative).

Original entry on oeis.org

63, 51, 20, 24, 44, 48, 112, 240, 608, 1552, 3120, 8144, 16304, 32624, 65264, 130544, 264928, 678448, 1356912, 4979232, 19424016, 58272480, 226593936, 763164288, 3467499840, 16339520448, 65370077568, 295266178368, 1223245608192, 6931725175296, 40582548986112

Offset: 1

N. J. A. Sloane, Dec 14 2003

nonn

A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.

Cf. A003415, A090636, A090637.

A090635(n, a=63)={if(n<0, vector(-n, n, if(n>1, a=A003415(a), a)), for(n=2, n, a=A003415(a)); a)}  \\ For n<0 return the vector a[1..-n]. - M. F. Hasler, Nov 27 2019

a(n+1) = A003415(a(n)), a(1) = 63. a(n) = 4*A129286(n-3) for n > 2. - M. F. Hasler, Nov 27 2019

More explicit name from M. F. Hasler, Nov 27 2019

A129285 a(n) = A129151(n) / 27.

Original entry on oeis.org

3, 4, 8, 20, 44, 92, 188, 380, 856, 2148, 5024, 17616, 58768, 176320, 755904, 3305920, 13885184, 69634816, 348174336, 2385273856, 14652403712, 102566830080, 849285738496, 6035962949632, 44017806979072, 308166534991872, 2380768960708608, 23410894780694528

Offset: 0

Reinhard Zumkeller, Apr 07 2007

nonn

Cf. A129284 (essentially the same), A129283 (n + n'), A129286, A051674, A129150, A090636 (trajectory of 15 under arithmetic derivative A003415).

a129285 n = a129151 n `div` 27  -- Reinhard Zumkeller, Nov 01 2013 [Moved here from A129284, first erroneous line deleted. - M. F. Hasler, Nov 27 2019]

a(n+1) = A129283(a(n)), a(0) = 3.
a(n) = A129284(n+1). - Eric M. Schmidt, Oct 22 2013
Thus a(n) = A129150(n+1) / 4 = A090636(n+3) / 4. - M. F. Hasler, Nov 27 2019

A090637 Trajectory of 28 under the map k -> A003415(k) (taking the arithmetic derivative).

Original entry on oeis.org

28, 32, 80, 176, 368, 752, 1520, 3424, 8592, 20096, 70464, 235072, 705280, 3023616, 13223680, 55540736, 278539264, 1392697344, 9541095424, 58609614848, 410267320320, 3397142953984, 24143851798528, 176071227916288, 1232666139967488, 9523075842834432

Offset: 1

N. J. A. Sloane, Dec 14 2003

nonn

Equals A090636 from a(2) = 32 = A090636(5) on. See there and A090635 for more.

			a(2) = 28' = (2^2*7)' = 28*(2/2 + 1/7) = 32.

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A090636, A090635.

a:= proc(n) option remember; `if`(n=1, 28, (t->
      t*add(i[2]/i[1], i=ifactors(t)[2]))(a(n-1)))
    end:
seq(a(n), n=1..31);  # Alois P. Heinz, Dec 02 2019

a[n_] := a[n] = If[n == 1, 28, Function[t, t*Sum[i[[2]]/i[[1]], {i, FactorInteger[t]}]][a[n-1]]];Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

PARI
```
a(n)=A090636(n,28)
```

Edited by M. F. Hasler, Nov 27 2019

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.

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A129150 The n-th arithmetic derivative of 2^3.

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A090635 Trajectory of 63 under the map k -> A003415(k) (taking the arithmetic derivative).

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A129285 a(n) = A129151(n) / 27.

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A090637 Trajectory of 28 under the map k -> A003415(k) (taking the arithmetic derivative).

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