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

Author

Michel Lagneau, Aug 24 2019

Keywords

• nonn

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

Examples

			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).
		

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000040, A000720, A006450, A027746, A027748

Contains A033845, A033849, A143207.

Programs

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

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 *)

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

Extensions

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