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A189553 Irregular triangle in which row n contains numbers x such that x'=n, where x' denotes the arithmetic derivative (A003415). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified March 22 17:25 EDT 2019. Contains 321422 sequences. (Running on oeis4.)