A140052 Indices m such that A114850(m)+A114850(k) is prime for some k<m.

6, 9, 9, 19, 20, 25, 33, 38, 40, 59, 69, 76, 99, 111, 126, 141, 147, 167, 188, 202, 211, 211, 220, 238, 263, 264, 279, 284, 297, 329, 336, 354, 407, 407, 407, 410, 426, 540, 568, 652, 683, 696, 769, 780, 948, 951

Offset

1,1

Comments

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).

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

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)."

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.

Links

Table of n, a(n) for n=1..46.

Formula

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

Example

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

Maple

? 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]", ")))

Crossrefs

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

Sequence in context: A307053 A100403 A066002 * A316072 A332439 A070598

Adjacent sequences: A140049 A140050 A140051 * A140053 A140054 A140055

Keyword

more,nonn

Author

Jonathan Vos Post, May 03 2008

Extensions

a(14)-a(46) from Donovan Johnson, Nov 11 2008

Status

approved