cp's OEIS Frontend

A173909 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.

%I A173909 #22 Jul 06 2019 17:20:20
%S A173909 3,5,7,9,10,15,17,18,20,24,29,32,39,42,47,55,57,62
%N A173909 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 A173909 From _Jon E. Schoenfield_, Apr 12 2014: (Start)
%C A173909 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 A173909 prime(80)  =  409:   91^318  +  318^91;
%C A173909 prime(84)  =  433:  111^322  +  322^111;
%C A173909 prime(87)  =  449:  214^235  +  235^214;
%C A173909 prime(94)  =  491:   20^471  +  471^20;
%C A173909 prime(129) =  727:   91^636  +  636^91;
%C A173909 prime(135) =  761:   98^663  +  663^98;
%C A173909 prime(136) =  769:  364^405  +  405^364;
%C A173909 prime(140) =  809:  365^444  +  444^365;
%C A173909 prime(142) =  821:   87^734  +  734^87;
%C A173909 prime(146) =  839:  329^510  +  510^329;
%C A173909 prime(149) =  859:  423^436  +  436^423;
%C A173909 prime(152) =  881:  291^590  +  590^291;
%C A173909 prime(158) =  929:  441^488  +  488^441;
%C A173909 prime(175) = 1039:  325^714  +  714^325;
%C A173909 prime(185) = 1103:  513^590  +  590^513;
%C A173909 prime(194) = 1181:  278^903  +  903^278;
%C A173909 prime(199) = 1217:   61^1156 + 1156^61;
%C A173909 prime(205) = 1259:  101^1158 + 1158^101;
%C A173909 prime(206) = 1277:  394^883  +  883^394;
%C A173909 prime(207) = 1279:  376^903  +  903^376;
%C A173909 prime(221) = 1381:  634^747  +  747^634;
%C A173909 prime(222) = 1399:  384^1015 + 1015^384;
%C A173909 prime(227) = 1433:  397^1036 + 1036^397. (End)
%e A173909 3 is in the sequence because 2^3 + 3^2 is prime and 2+3 = 5 = 3rd prime;
%e A173909 5 is in the sequence because 2^9 + 9^2 is prime and 2+9 = 11 = 5th prime;
%e A173909 7 is in the sequence because 2^15 + 15^2 is prime and 2+15 = 17 = 7th prime;
%e A173909 9 is in the sequence because 2^21 + 21^2 is prime and 2+21 = 23 = 9th prime;
%e A173909 10 is in the sequence because 5^24 + 24^5 is prime and 5+24 = 29 = 10th prime.
%Y A173909 Cf. A061119, A094133, A162488, A162490.
%K A173909 nonn,more
%O A173909 1,1
%A A173909 _Juri-Stepan Gerasimov_, Mar 02 2010
%E A173909 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