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A350069 Semiprimes k such that 1+(2^(1+A243055(k))) is a Fermat prime, where A243055(k) gives the difference between the indices of the smallest and the largest prime divisor of k.

%I A350069 #11 Jan 29 2022 20:16:20
%S A350069 4,6,9,14,15,25,33,35,38,49,65,69,77,106,119,121,143,145,169,177,209,
%T A350069 217,221,289,299,305,323,361,407,437,469,493,529,533,589,667,731,781,
%U A350069 841,851,893,899,949,961,1147,1189,1219,1333,1343,1369,1517,1577,1681,1711,1739,1763,1849,1891,2021,2047,2173,2209,2479
%N A350069 Semiprimes k such that 1+(2^(1+A243055(k))) is a Fermat prime, where A243055(k) gives the difference between the indices of the smallest and the largest prime divisor of k.
%H A350069 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A350069 9 is a semiprime (9 = 3*3), and as the difference between the indices of the smallest (3) and the largest prime (3) dividing 9 is 0, we have 1+(2^(1+A243055(k))) = 3, which is in A019434, and therefore 9 is included in this sequence, like all squares of primes (A001248).
%e A350069 177 = 3 * 59 = prime(2) * prime(17), therefore A243055(177) = 17-2 = 15, and as 1+(2^16) = 65537 is also in A019434, 177 is included in this sequence.
%o A350069 (PARI)
%o A350069 A209229(n) = (n && !bitand(n,n-1));
%o A350069 A243055(n) = if(1==n,0,my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (primepi(gpf)-primepi(lpf)));
%o A350069 isA350069(n) = if(2!=bigomega(n),0,my(d=1+A243055(n)); (A209229(d) && isprime(1+(2^d))));
%Y A350069 Positions of ones in A342651.
%Y A350069 Cf. A019434, A209229, A243055, A334101.
%Y A350069 Subsequence of A001358. A001248 is a subsequence.
%K A350069 nonn
%O A350069 1,1
%A A350069 _Antti Karttunen_, Jan 29 2022