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 A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes. 3
 4, 9, 10, 25, 49, 65, 289 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Semiprimes both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1 and where the semiprime is itself of the form (2^K)*(3^L)+1. No more under 10^50; what is the next element of this sequence? No more terms <= 10^100. - Robert Israel, Mar 10 2017 This sequence is complete, see Links. - Charlie Neder, Feb 04 2019 LINKS Caldwell, C., "Pierpont primes." primeform posting, Oct 25, 2005. Charlie Neder, Proof of the completeness of this sequence Eric Weisstein's World of Mathematics, Pierpont Prime Eric Weisstein's World of Mathematics, Semiprime FORMULA {a(n)} = intersection of A113432 and A113433. {a(n)} = Semiprimes A001358 of the form (2^K)*(3^L)+1 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {integers P such that, for nonnegative integers I, J, K, L, m, n there is a solution to (2^I)*(3^J)+1 = [(2^K)*(3^L)+1]*[(2^m)*(3^n)+1] where both [(2^K)*(3^L)+1] and [(2^m)*(3^n)+1] are prime}. EXAMPLE a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^0)*(3^0)+1] = (2^0)*(3^1)+1. a(2) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = (2^3)*(3^0)+1. a(3) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^0)*(3^2)+1. a(4) = 25 = 5^2 = [(2^2)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^3)*(3^1)+1. a(5) = 49 = 7^2 = [(2^1)*(3^1)+1]*[(2^1)*(3^1)+1] = (2^4)*(3^1)+1. a(6) = 65 = 5*13 = [(2^2)*(3^0)+1]*[(2^2)*(3^1)+1] = (2^6)*(3^0)+1. a(7) = 289 = 17^2 = [(2^4)*(3^0)+1]*[(2^4)*(3^0)+1] = (2^5)*(3^2)+1. MAPLE N:= 10^100: # to get all terms <= N PP:= select(isprime, {seq(seq(1+2^i*3^j, i=0..ilog2((N-1)/3^j)), j=0..floor(log(N-1)))}): SP:= select(t -> t <= N and t = 1+2^padic:-ordp(t-1, 2)*3^padic:-ordp(t-1, 3), [seq(seq(PP[i]*PP[j], j=1..i), i=1..nops(PP))]): sort(convert(SP, list)); # Robert Israel, Mar 10 2017 CROSSREFS Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113433. Sequence in context: A191905 A113432 A129830 * A236024 A141395 A121215 Adjacent sequences:  A113431 A113432 A113433 * A113435 A113436 A113437 KEYWORD nonn,fini,full AUTHOR Jonathan Vos Post, Nov 01 2005 STATUS approved

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Last modified August 20 14:06 EDT 2019. Contains 326152 sequences. (Running on oeis4.)