A360357 - OEIS

%I #11 Feb 06 2023 01:28:13

%S 1,7,13,28,37,46,52,73,91,97,103,106,112,148,151,172,181,193,196,202,

%T 223,226,232,256,262,292,298,301,316,337,343,346,361,376,388,397,427,

%U 448,457,463,466,478,487,502,511,523,541,556,568,592,601,607,613,622,631

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

%C 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).

%C 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).

%H Amiram Eldar, <a href="/A360357/b360357.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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

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

%o lista(nmax) = {my(q1 = is(1), q2); for(n = 2, nmax, q2 = is(n); if(q1 && q2, print1(n-1, ", ")); q1 = q2); }

%Y Cf. A000668, A006450, A059305, A320628.

%K nonn

%O 1,2

%A _Amiram Eldar_, Feb 04 2023