A140052 - OEIS

%I #8 Apr 04 2015 10:01:57

%S 6,9,9,19,20,25,33,38,40,59,69,76,99,111,126,141,147,167,188,202,211,

%T 211,220,238,263,264,279,284,297,329,336,354,407,407,407,410,426,540,

%U 568,652,683,696,769,780,948,951

%N Indices m such that A114850(m)+A114850(k) is prime for some k<m.

%C The associated primes b(n), which grow too quickly for many to be given as a sequence themselves, are {primes of the form A114850(a) + A114850(b)} = {primes of the form A114850(a) + A114850(b)} and begin as follows. b(1) = 437893890380859631 = 256 + 437893890380859375 = 4^4 + 15^15 = semiprime(1)^semiprime(1) + semiprime(6)^semiprime(6).

%C b(2) = 88817841970012523233890533447265881 = 256 + 88817841970012523233890533447265625 = 4^4 + 25^25 = semiprime(1)^semiprime(1) + semiprime(9)^semiprime(9).

%C b(3) = 46656 + 88817841970012523233890533447265625 = 6^6 + 24^25 = semiprime(2)^semiprime(2) + semiprime(9)^semiprime(9). This is to A068145 "Primes of the form a^a + b^b" as A001358 semiprimes is to A000040 primes; and as A114850 "(n-th semiprime)^(n-th semiprime)" is to A051674 "(n-th prime)^(n-th prime)."

%C _M. F. Hasler_ gave the present definition which allows us to list merely the indices, which in the 3 examples above, are [6, 1],[9, 1],[9, 2]. The first 13 [m,k] value pairs are (as found by _M. F. Hasler_ as an extension) are [6, 1], [9, 1], [9, 2], [19, 5], [20, 8], [25, 7], [33, 11], [38, 6], [40, 33], [59, 14], [69, 62], [76, 57], [99, 22]. Hence our sequence begins a(1) = 6, a(2) = 9, a(3) = 9. For the sequence of corresponding k values {1, 1, 2, 5, 8, ...}, see A140053.

%F A001358(a(n))^A001358(a(n)) + A001358(A140053(n))^A001358(A140053(n)) is prime.

%e a(1) = 6 because semiprime(6)^semiprime(6) + semiprime(1)^semiprime(1) = 15^15 + 4^4 = 437893890380859375 + 256 = 437893890380859631 is prime.

%p ? t=0;A001358=vector(100,i,until(bigomega(t++)==2,);t); ? for(i=1,#A001358, for(j=1,i-1, ispseudoprime(A001358[i]^A001358[i]+A001358[j]^A001358[j]) | next; print1([i,j]",")))

%Y Cf. A000040, A001358, A051674, A068145, A114850, A137701, A140053.

%K more,nonn

%O 1,1

%A _Jonathan Vos Post_, May 03 2008

%E a(14)-a(46) from _Donovan Johnson_, Nov 11 2008