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 A231557 Least positive integer k <= n such that 2^k + (n - k) is prime, or 0 if such an integer k does not exist. 14
 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 6, 3, 2, 1, 2, 1, 4, 5, 2, 1, 8, 3, 4, 3, 2, 1, 2, 1, 4, 3, 8, 5, 2, 1, 10, 3, 2, 1, 2, 1, 6, 5, 2, 1, 4, 3, 4, 11, 2, 1, 20, 3, 4, 3, 2, 1, 2, 1, 4, 3, 8, 13, 2, 1, 4, 3, 2, 1, 2, 1, 6, 3, 12, 5, 2, 1, 6, 5, 2, 1, 8, 3, 4, 5, 2, 1, 4, 7, 4, 3, 6, 11, 2, 1, 4, 3, 2, 1 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000040, A000079, A231201, A231555, A231725. Sequence in context: A276976 A135545 A123317 * A171453 A285707 A164879 Adjacent sequences:  A231554 A231555 A231556 * A231558 A231559 A231560 KEYWORD nonn AUTHOR Zhi-Wei Sun, Nov 11 2013 STATUS approved

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Last modified October 18 05:22 EDT 2019. Contains 328146 sequences. (Running on oeis4.)