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

1,2

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).

Links

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

Example

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

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

Prog

(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); }

Crossrefs

Cf. A000668, A006450, A059305, A320628.

Sequence in context: A146646 A096194 A160889 * A283650 A045463 A082221

Adjacent sequences: A360354 A360355 A360356 * A360358 A360359 A360360

Keyword

nonn

Author

Amiram Eldar, Feb 04 2023

Status

approved