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

Author

Vladimir Shevelev, Jun 25 2011

Keywords

• nonn

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)

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.

Programs

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 *)

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

Formula

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

Extensions

More terms from Olivier Gérard, Jul 04 2011

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