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Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.
3

%I #25 Feb 17 2015 00:15:45

%S 72,738,2758,16777232,1073741856,282429536508,95367431640650,

%T 150094635296999148,221073919720733357899812,

%U 311973482284542371301330321821976098,1329227995784915872903807060280344640,85070591730234615865843651857942052992

%N Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.

%C Intersection of A253913 and A253775.

%H Robert G. Wilson v, <a href="/A253917/b253917.txt">Table of n, a(n) for n = 1..17</a>

%H Robert G. Wilson v, <a href="/A253917/a253917.txt">51 identified terms with no assurance that these are the only terms less than 10^1000.</a>

%e 72 = 2^6+2+6 = 8^2+8,

%e 738 = 3^6+3+6 = 9^3+9,

%e 2758 = 52^2+52+2 = 14^3+14,

%e 16777232 = 4^12+4+12 = 8^8+8,

%e 1073741856 = 2^30+2+30 = 32^6+32,

%e 282429536508 = 3^24+3+24 = 27^8+27,

%e 95367431640650 = 5^20+5+20 = 25^10+25,

%e 150094635296999148 = 9^18+9+18 = 27^12+27,

%e 221073919720733357899812 = 6^30+6+30 = 30^15+36,

%e 311973482284542371301330321821976098 = 7^42+7+42 = 49^21+49,

%e 1329227995784915872903807060280344640 = 4^60+4+60 = 64^20+64,

%e 85070591730234615865843651857942052992 = 2^126+2+126 = 128^18+128,

%e etc. - _Robert G. Wilson v_, Jan 19 2015

%t f[n_] := Block[{t = Transpose@ Flatten[ Table[{m^k + m, m^k + m + k}, {m, 2, Floor@ Sqrt[2^n]}, {k, Floor@ Log[m, 2^(n - 1)] + 1, Floor@ Log[m, 2^n]}], 1]}, Intersection[ t[[1]], t[[2]]]]; f[1] = {}; Array[f, 50] // Flatten (* _Robert G. Wilson v_, Jan 19 2015 *)

%Y Cf. A253913, A253775, A253914, A253916.

%K nonn,more

%O 1,1

%A _Alex Ratushnyak_, Jan 18 2015

%E a(7)-a(12) from _Robert G. Wilson v_, Jan 19 2015