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%I #27 Apr 06 2017 04:37:46
%S 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,
%T 43,46,47,49,53,57,58,59,61,62,66,67,70,71,73,78,79,82,83,85,89,93,94,
%U 97,101,103,105,107,109,113,114,118,121,127,129,130
%N Polynomial-like numbers: numbers whose sequence of number-derivatives (A003415) is monotonically decreasing.
%C 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.
%C 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).
%C 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.
%C 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.
%H Alois P. Heinz, <a href="/A192189/b192189.txt">Table of n, a(n) for n = 1..1000</a>
%e 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.
%e 49 is in the sequence: A003415^(i)(49)|_{i=0..6} = 49, 14, 9, 6, 5, 1, 0.
%p d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
%p a:= proc(n) option remember; local i, j, k;
%p for k from 1 +`if`(n=1, 0, a(n-1)) do
%p i, j:= d(k), k;
%p while i<>0 and i<j do i, j:= d(i), i od;
%p if i=0 then return k fi
%p od
%p end:
%p seq(a(n), n=1..80); # _Alois P. Heinz_, Jul 22 2011
%t d[n_] := If[n < 2, 0, n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]];
%t 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]]];
%t Array[a, 80] (* _Jean-François Alcover_, Apr 06 2017, after _Alois P. Heinz_ *)
%Y Cf. A003415, A038554, A192082.
%K nonn
%O 1,2
%A _Vladimir Shevelev_, Jun 25 2011
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