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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}).

%I A309944 #26 Sep 08 2022 08:46:22
%S A309944 6,12,15,18,24,30,36,45,48,54,55,60,72,75,90,96,108,119,120,135,144,
%T A309944 150,162,165,180,192,216,225,240,270,275,288,300,324,330,341,360,375,
%U A309944 384,405,432,450,480,486,495,533,540,576,600,605,648,660,675,720,750,768
%N 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}).
%C A309944 Numbers m such that for all k, d(k) = prime(d(k-1)), where d(k) is the k-th prime factor of m.
%C A309944 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, ...
%C A309944 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
%H A309944 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A309944 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).
%e A309944 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).
%p A309944 with(numtheory):nn:=10^3:
%p A309944 for n from 1 to nn do:
%p A309944 d:=factorset(n):n0:=nops(d):it:=0:
%p A309944   if n0>1
%p A309944   then
%p A309944   for i from 2 to n0 do :
%p A309944    if d[i]=ithprime(d[i-1])
%p A309944     then
%p A309944     it:=it+1:
%p A309944     else fi:
%p A309944    od:
%p A309944     if it=n0-1
%p A309944     then
%p A309944     printf(`%d, `,n):
%p A309944     else fi:fi:
%p A309944 od:
%t A309944 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 *)
%o A309944 (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
%o A309944 (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
%Y A309944 Cf. A000040, A000720, A006450, A027746, A027748
%Y A309944 Contains A033845, A033849, A143207.
%K A309944 nonn
%O A309944 1,1
%A A309944 _Michel Lagneau_, Aug 24 2019
%E A309944 Edited by _N. J. A. Sloane_, Oct 05 2019, using definition suggested by _Antti Karttunen_, Aug 24 2019