A360357 Numbers k such that k and k+1 are both products of primes of nonprime index (A320628).

1, 7, 13, 28, 37, 46, 52, 73, 91, 97, 103, 106, 112, 148, 151, 172, 181, 193, 196, 202, 223, 226, 232, 256, 262, 292, 298, 301, 316, 337, 343, 346, 361, 376, 388, 397, 427, 448, 457, 463, 466, 478, 487, 502, 511, 523, 541, 556, 568, 592, 601, 607, 613, 622, 631

Offset: 1

Author

Amiram Eldar, Feb 04 2023

Keywords

• nonn

Comments

There are no 3 consecutive integers that are products of primes of nonprime index since 1 out of 3 consecutive integers is divisible by 3 which is a prime-indexed prime (A006450).

If a Mersenne prime (A000668) is a prime of nonprime index, then it is in this sequence. Of the first 10 Mersenne primes 6 are in this in sequence: A000668(k) for k = 2, 5, 7, 8, 9, 10 (see A059305).

Examples

			7 = prime(4) is a term since 4 is nonprime, 7 + 1 = 8 = prime(1)^3, and 1 is also nonprime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000668, A006450, A059305, A320628.

Programs

Mathematica

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], ! PrimeQ[PrimePi[#]] &]; seq = {}; q1 = q[1]; n = 2; c = 0; While[c < 55, q2 = q[n]; If[q1 && q2, c++; AppendTo[seq, n - 1]]; q1 = q2; n++]; seq

PARI

is(n) = {my(p = factor(n)[,1]); for(i = 1, #p, if(isprime(primepi(p[i])), return(0))); 1;}
lista(nmax) = {my(q1 = is(1), q2); for(n = 2, nmax, q2 = is(n); if(q1 && q2, print1(n-1, ", ")); q1 = q2); }