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A360357
Numbers k such that k and k+1 are both products of primes of nonprime index (A320628).
2
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Feb 04 2023
STATUS
approved