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Starts of runs of at least 4 consecutive odd numbers whose prime factors are all prime-indexed primes.
2

%I #9 Sep 17 2022 14:21:22

%S 121,1199,1409,16141,56699,474529,695235,1780713,1917997,6196985,

%T 7209817,7559673,8084871,11403485,14409147,22405711,22608861,23261179,

%U 25803873,27844653,28729833,31126321,35664449,43527369,44425215,48690429,62579001,63706967,66780601

%N Starts of runs of at least 4 consecutive odd numbers whose prime factors are all prime-indexed primes.

%C There are no runs of 7 consecutive odd numbers with this property, since in every run of 7 consecutive odd numbers one is divisible by 7 which is not a prime-indexed prime.

%C Are there such runs of 5 consecutive odd numbers? There are none below 6.6*10^7.

%e 121 = 11^2 is a term since 123 = 3 * 41, 125 = 5^3, and 3 = prime(2), 5 = prime(3), 11 = prime(5), 41 = prime(13) and 127 = prime(31) are all prime-indexed primes.

%t q[n_] := AllTrue[FactorInteger[n][[;; , 1]], PrimeQ[PrimePi[#]] &]; q[1] = True; v = q /@ {1, 3, 5, 7}; seq = {}; Do[If[And @@ v, AppendTo[seq, k - 8]]; v = Join[Rest[v], {q[k]}], {k, 9, 10^6, 2}]; seq

%o (PARI) is(n) = {my(p = factor(n)[,1]); for(i = 1, #p, if(!isprime(primepi(p[i])), return(0))); return(1)};

%o v = vector(4); forstep(k = 3, 9, 2, v[(k-1)/2] = is(k));

%o forstep(k=11, 1e8, 2, q = is(k); v = concat(vecextract(v,"^1"),q); if(v[1]&&v[2]&&v[3]&&v[4], print1(k-6,", ")))

%Y Cf. A006450, A076610.

%Y Subsequence of A357167 and A357168.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 16 2022