A189553 Irregular triangle in which row n contains numbers x such that x'=n, where x' denotes the arithmetic derivative (A003415).

4, 6, 9, 10, 15, 14, 21, 25, 8, 35, 22, 33, 49, 26, 12, 39, 55, 65, 77, 34, 51, 91, 18, 38, 57, 85, 121, 20, 95, 119, 143, 46, 69, 133, 169, 27, 115, 187, 161, 209, 221, 30, 58, 16, 28, 87, 247, 62, 93, 145, 253, 289, 155, 203, 299, 323, 217, 361, 45, 74

Offset

4,1

Comments

Row 0 contains 0 and 1. Row 1 contains all primes. Rows 2 and 3 are empty. Hence, we start this table with row 4. The length of row n is A099302(n). The first term in row n is A098699(n). The last term is A099303(n). Row n is the set I(n) in the paper by Ufnarovski and Ahlander. They show that all terms in row n are <= (n/2)^2. The upper bound is attained when n = 2p, where p is a prime.

References

See A003415.

Links

T. D. Noe, Rows n = 4..1000, flattened

Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Example

The triangle begins
4
6
9
10
15
14
21, 25
none
8, 35
22
33, 49
26
12, 39, 55

Mathematica

dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; nn = 100; d = Array[dn, (nn/2)^2]; Table[Flatten[Position[d, n]], {n, 4, nn}]

Crossrefs

Cf. A003415, A098699, A099302, A099303.

Sequence in context: A132435 A108631 A200677 * A189482 A099303 A243485

Adjacent sequences: A189550 A189551 A189552 * A189554 A189555 A189556

Keyword

nonn,tabf

Author

T. D. Noe, Apr 23 2011

Status

approved