OFFSET
1,3
COMMENTS
Conjecture: a(n) > 0 for all n > 0.
See also part (i) of the conjecture in A231201.
We have computed a(n) for all n up to 2*10^6 except for n = 1657977. Here are some relatively large values of a(n): a(421801) = 149536 (the author found that 2^{149536} + 421801 - 149536 is prime, and then his friend Qing-Hu Hou verified that 2^k + 421801 - k is composite for each integer 0 < k < 149536), a(740608) = 25487, a(768518) = 77039, a(1042198) = 31357, a(1235105) = 21652, a(1253763) = 39018, a(1310106) = 55609, a(1346013) = 33806, a(1410711) = 45336, a(1497243) = 37826, a(1549802) = 21225, a(1555268) = 43253, a(1674605) = 28306, a(1959553) = 40428.
Now we find that a(1657977) = 205494. The prime 2^205494 + (1657977-205494) has 61860 decimal digits. - Zhi-Wei Sun, Aug 30 2015
We have found that a(n) > 0 for all n = 1..7292138. For example, a(5120132) = 250851, and the prime 2^250851 + 4869281 has 75514 decimal digits. - Zhi-Wei Sun, Nov 16 2015
We have verified that a(n) > 0 for all n = 1..10^7. For example, a(7292139) = 218702 and 2^218702 + (7292139-218702) is a prime of 65836 decimal digits; also a(9302003) = 311468 and the prime 2^311468 + (9302003-311468) has 93762 decimal digits. - Zhi-Wei Sun, Jul 28 2016
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Write n = k + m with 2^k + m prime, a message to Number Theory List, Nov. 16, 2013.
Zhi-Wei Sun, On a^n+bn modulo m, arXiv:1312.1166 [math.NT], 2013-2014.
EXAMPLE
a(1) = 1 since 2^1 + (1-1) = 2 is prime.
a(2) = 1 since 2^1 + (2-1) = 3 is prime.
a(3) = 2 since 2^1 + (3-1) = 4 is not prime, but 2^2 + (3-2) = 5 is prime.
MATHEMATICA
Do[Do[If[PrimeQ[2^x+n-x], Print[n, " ", x]; Goto[aa]], {x, 1, n}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 100}]
PROG
(PARI) a(n) = {for (k = 1, n, if (isprime(2^k+n-k), return (k)); ); return (0); } \\ Michel Marcus, Nov 11 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 11 2013
STATUS
approved