A309944 Numbers m such that if m = p_1^e_1 * ... * p_k^e_k, where p_1 < ... < p_k are primes, then for all i < k, p_i = A000720(p_{i+1}).

6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 55, 60, 72, 75, 90, 96, 108, 119, 120, 135, 144, 150, 162, 165, 180, 192, 216, 225, 240, 270, 275, 288, 300, 324, 330, 341, 360, 375, 384, 405, 432, 450, 480, 486, 495, 533, 540, 576, 600, 605, 648, 660, 675, 720, 750, 768

Offset

1,1

Comments

Numbers m such that for all k, d(k) = prime(d(k-1)), where d(k) is the k-th prime factor of m.

The primitive subsequence b(k), k = 1, 2, ... begins with 6, 15, 30, 55, 110, 165, 330, 341, 533, ... because if d(i) is the i-th prime factor of b(k), so b(k)*d(i)^m is in the sequence, m = 0, 1, 2, ...

Numbers m such that if m = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then for all i > 1, p_i = A000040(p_{i-1}). - Antti Karttunen, Aug 24 2019

Links

Table of n, a(n) for n=1..56.

Index entries for sequences computed from indices in prime factorization

Example

330 is in the sequence because the prime factors are {2, 3, 5, 11} with 3 = prime(2), 5 = prime(3) and 11 = prime(5).
1299210 is in the sequence because the prime factors are {2, 3, 5, 11, 31, 127} with 3 = prime(2), 5 = prime(3), 11 = prime(5), 31 = prime(11) and 127 = prime(31).

Maple

with(numtheory):nn:=10^3:
for n from 1 to nn do:
d:=factorset(n):n0:=nops(d):it:=0:
  if n0>1
  then
  for i from 2 to n0 do :
   if d[i]=ithprime(d[i-1])
    then
    it:=it+1:
    else fi:
   od:
    if it=n0-1
    then
    printf(`%d, `, n):
    else fi:fi:
od:

Mathematica

aQ[n_] := (m = Length[(p = FactorInteger[n][[;; , 1]])]) > 1 && NestList[Prime@# &, p[[1]], m - 1] == p; Select[Range[770], aQ] (* Amiram Eldar, Aug 24 2019 *)

Prog

(Magma) sol:=[]; s:=1; for m in [2..1000] do v:=PrimeDivisors(m);  if #v ge 2 then nr:=0; for k in [2..#v] do  if v[k] eq NthPrime(v[k-1])  then nr:=nr+1;  end if; end for; if nr eq #v-1 then sol[s]:=m; s:=s+1; end if; end if; end for;  sol; // Marius A. Burtea, Aug 24 2019
(PARI) isok(m) = {my(f=factor(m)[, 1]~); if (#f < 2, return(0)); for (i=2, #f, if (f[i] != prime(f[i-1]), return (0)); ); return (1); } \\ Michel Marcus, Aug 25 2019

Crossrefs

Cf. A000040, A000720, A006450, A027746, A027748

Contains A033845, A033849, A143207.

Sequence in context: A104210 A356736 A066312 * A212308 A089341 A256617

Adjacent sequences: A309941 A309942 A309943 * A309945 A309946 A309947

Keyword

nonn

Author

Michel Lagneau, Aug 24 2019

Extensions

Edited by N. J. A. Sloane, Oct 05 2019, using definition suggested by Antti Karttunen, Aug 24 2019

Status

approved