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

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

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

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], ! PrimeQ[PrimePi[#]] &]; Select[Range[1000], 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

allocated for Amiram EldarNumbers 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

1,2

There are no 3 consecutive integers that are products of primes of nonprime index since one 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).

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

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], ! PrimeQ[PrimePi[#]] &]; Select[Range[1000], q[#] && q[# + 1] &]

(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); }

Cf. A000668, A006450, A059305, A320628.

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Amiram Eldar, Feb 04 2023

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