A189803 Composite numbers n such that n'' = n'-1 where n' and n'' are the first and the second arithmetic derivative of n.

9, 185, 341, 377, 437, 9005, 30413, 33953, 41009, 51533, 82673, 92909, 103073, 126509, 143009, 165773, 181793, 184973, 191309, 228653, 231713, 246893, 291233, 311309, 316973, 319793, 329357, 353009, 358433, 374513, 398093, 405809, 431009, 460193, 467309

Offset: 1

Author

Giorgio Balzarotti, Apr 27 2011

Keywords

• nonn

Comments

The sequence A189710 (n"=n'-1) includes all prime numbers because p'=1 and p" = 0. Composite numbers are not very frequent.

Are all terms semiprimes? These terms appear to be p*q such that p+q is a term in A054377, which has solutions to the equation n' = n-1. - T. D. Noe, Apr 27 2011

Examples

			9' = 6, 9''= 6'= 5, 9" = 9'- 1 -> 9 is in the sequence.
		

Crossrefs

Cf. A002808, A003415, A054377, A068346, A189710.

Programs

PARI

ader(n) = my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1])); \\ A003415
isok(k) = if (!isprime(k), my(d=ader(k)); ader(d) == d - 1); \\ Michel Marcus, Mar 13 2023