A357169 Starts of runs of at least 4 consecutive odd numbers whose prime factors are all prime-indexed primes.

121, 1199, 1409, 16141, 56699, 474529, 695235, 1780713, 1917997, 6196985, 7209817, 7559673, 8084871, 11403485, 14409147, 22405711, 22608861, 23261179, 25803873, 27844653, 28729833, 31126321, 35664449, 43527369, 44425215, 48690429, 62579001, 63706967, 66780601

Offset

1,1

Comments

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.

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

Links

Table of n, a(n) for n=1..29.

Example

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.

Mathematica

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

Prog

(PARI) is(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(!isprime(primepi(p[i])), return(0))); return(1)};
v = vector(4); forstep(k = 3, 9, 2, v[(k-1)/2] = is(k));
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, ", ")))

Crossrefs

Cf. A006450, A076610.

Subsequence of A357167 and A357168.

Sequence in context: A208261 A036928 A088171 * A254970 A036308 A066444

Adjacent sequences: A357166 A357167 A357168 * A357170 A357171 A357172

Keyword

nonn

Author

Amiram Eldar, Sep 16 2022

Status

approved