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%I #22 Jul 06 2019 17:20:20
%S 3,5,7,9,10,15,17,18,20,24,29,32,39,42,47,55,57,62
%N Numbers n such that prime(n) can be expressed as x+y in at least one way such that x^y + y^x is prime and 1 < x <= y.
%C From _Jon E. Schoenfield_, Apr 12 2014: (Start)
%C All terms through 62 (as well as the term 83, which is in the sequence, but might not be next) were confirmed as having a corresponding prime expression of the form x^y + y^x using the online Magma Calculator. The next terms after 62 are probably 80, 83, 84, 87, 94, 129, 135, 136, 140, 142, 146, 149, 152, 158, 175, 185, 194, 199, 205, 206, 207, 221, 222, 227; these are the only values of n in 62 < n <= 236 for which at least one pair (x,y) yields a value of x^y + y^x that is a probable prime. Of these (at least probable) terms, 83 is definitely in the sequence (as 9^422 + 422^9 is definitely prime, and 9+422=431=prime(83)); for the rest, the probably-prime x^y + y^x with the smallest x (there may be more than one) is as follows:
%C prime(80) = 409: 91^318 + 318^91;
%C prime(84) = 433: 111^322 + 322^111;
%C prime(87) = 449: 214^235 + 235^214;
%C prime(94) = 491: 20^471 + 471^20;
%C prime(129) = 727: 91^636 + 636^91;
%C prime(135) = 761: 98^663 + 663^98;
%C prime(136) = 769: 364^405 + 405^364;
%C prime(140) = 809: 365^444 + 444^365;
%C prime(142) = 821: 87^734 + 734^87;
%C prime(146) = 839: 329^510 + 510^329;
%C prime(149) = 859: 423^436 + 436^423;
%C prime(152) = 881: 291^590 + 590^291;
%C prime(158) = 929: 441^488 + 488^441;
%C prime(175) = 1039: 325^714 + 714^325;
%C prime(185) = 1103: 513^590 + 590^513;
%C prime(194) = 1181: 278^903 + 903^278;
%C prime(199) = 1217: 61^1156 + 1156^61;
%C prime(205) = 1259: 101^1158 + 1158^101;
%C prime(206) = 1277: 394^883 + 883^394;
%C prime(207) = 1279: 376^903 + 903^376;
%C prime(221) = 1381: 634^747 + 747^634;
%C prime(222) = 1399: 384^1015 + 1015^384;
%C prime(227) = 1433: 397^1036 + 1036^397. (End)
%e 3 is in the sequence because 2^3 + 3^2 is prime and 2+3 = 5 = 3rd prime;
%e 5 is in the sequence because 2^9 + 9^2 is prime and 2+9 = 11 = 5th prime;
%e 7 is in the sequence because 2^15 + 15^2 is prime and 2+15 = 17 = 7th prime;
%e 9 is in the sequence because 2^21 + 21^2 is prime and 2+21 = 23 = 9th prime;
%e 10 is in the sequence because 5^24 + 24^5 is prime and 5+24 = 29 = 10th prime.
%Y Cf. A061119, A094133, A162488, A162490.
%K nonn,more
%O 1,1
%A _Juri-Stepan Gerasimov_, Mar 02 2010
%E Constraint "0<x<=y" in definition changed to "1<x<=y" (and related edits made) by _Jon E. Schoenfield_ (after comments from _R. J. Mathar_ regarding missing terms and from _Wolfdieter Lang_ noting that the existing definition would make this sequence identical to A000027), Apr 12 2014
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