A192189 Polynomial-like numbers: numbers whose sequence of number-derivatives (A003415) is monotonically decreasing.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 21, 22, 23, 25, 29, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 66, 67, 70, 71, 73, 78, 79, 82, 83, 85, 89, 93, 94, 97, 101, 103, 105, 107, 109, 113, 114, 118, 121, 127, 129, 130

Offset: 1

Author

Vladimir Shevelev, Jun 25 2011

Keywords

• nonn

Comments

Every prime is in the sequence. If n==0 (mod 4), then it is not in the sequence. Moreover, if, for a prime p, n==0 (mod p^p), then n is not in the sequence. Indeed, if n=(p^p)*k, then n'=(p^p)'*k+p^p*k'=p^p(k+k')>=n, analogously, n''>=n', etc.

Conjecture. For every sufficiently large greater q of twin primes, the sequence contains infinite increasing sequence {s_n} of semiprimes beginning with 2*(q-2), such that (s_n)'=s_(n-1).

This conjecture is true, if 1) there exist infinitely many twin primes; 2) there exists n_0, such that for every prime p>n_0, number 2*p is sum of two primes r,t, for which r*t-2 is prime.

Proof. Let q>=n_0. Put s_1=2(q-2). By the condition, 2(q-2)=r+t, such that r*t-2 is prime. Put s_2=r*t and s_3=2(r*t-2). Then (s_3)'=2'*(r*t-2)+2*(r*t-2)'=s_2; (s_2)'=r+t=s_1. Continuing this process, we get an infinite sequence of semiprimes and every semiprime is a number whose sequence of number-derivatives is monotonically decreasing.

Examples

			Let n=50. We have A003415(50) = 45, A003415(45) = 39, A003415(39) = 16 but a(16) = 32. Thus 50 is not in the sequence.
49 is in the sequence: A003415^(i)(49)|_{i=0..6} = 49, 14, 9, 6, 5, 1, 0.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003415, A038554, A192082.

Programs

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember; local i, j, k;
      for k from 1 +`if`(n=1, 0, a(n-1)) do
        i, j:= d(k), k;
        while i<>0 and iAlois P. Heinz, Jul 22 2011

Mathematica

d[n_] := If[n < 2, 0, n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]];
a[n_] := a[n] = For[k = 1+If[n == 1, 0, a[n-1]], True, k++, {i, j} = {d[k], k}; While[i != 0 && i < j, {i, j} = {d[i], i}]; If [i == 0, Return[k]]];
Array[a, 80] (* Jean-François Alcover, Apr 06 2017, after Alois P. Heinz *)