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A192189 Polynomial-like numbers: numbers whose sequence of number-derivatives (A003415) is monotonically decreasing. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

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

EXAMPLE

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.

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 i<j do i, j:= d(i), i od;

        if i=0 then return k fi

      od

    end:

seq(a(n), n=1..80);  # Alois 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-Fran├žois Alcover, Apr 06 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A003415, A038554, A192082.

Sequence in context: A099308 A074235 A325366 * A285375 A321372 A001948

Adjacent sequences:  A192186 A192187 A192188 * A192190 A192191 A192192

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Jun 25 2011

STATUS

approved

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Last modified April 19 17:46 EDT 2021. Contains 343117 sequences. (Running on oeis4.)