A192192 Numbers whose second arithmetic derivative (A068346) is prime; Polynomial-like numbers of degree 3.

9, 21, 25, 57, 85, 93, 121, 126, 145, 161, 185, 201, 206, 209, 221, 237, 242, 253, 265, 289, 305, 315, 326, 333, 341, 365, 369, 377, 381, 413, 417, 437, 453, 458, 490, 495, 497, 517, 537, 542, 545, 565, 566, 575, 578, 597, 605, 633, 637, 638, 649, 666, 685

Offset

1,1

Comments

The fourth A003415-iteration of a(n) is the first to be 0.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10001 (the first 1000 terms from T. D. Noe)

Formula

For all n, A327969(a(n)) <= 4. - Antti Karttunen, Oct 19 2019

Mathematica

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]]]]]; Select[Range[1000], dn[dn[dn[#]]] == 1&] (* T. D. Noe, Mar 07 2013 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA192192(n) = isprime(A003415(A003415(n))); \\ Antti Karttunen, Oct 19 2019

Crossrefs

Cf. A003415, A038554, A068346, A192082, A192189, A192190, A327969, A328240.

Cf. A157037, A328239 (the first and third derivative is prime).

Subsequence of following sequences: A328234, A328244, A328246.

Sequence in context: A273202 A338909 A173456 * A053795 A100490 A173250

Adjacent sequences: A192189 A192190 A192191 * A192193 A192194 A192195

Keyword

nonn

Author

Vladimir Shevelev, Jun 25 2011

Extensions

More terms from Olivier Gérard, Jul 04 2011

New primary definition added to the name by Antti Karttunen, Oct 19 2019

Status

approved