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A253917 Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.

%I A253917 #29 Aug 08 2025 14:20:24
%S A253917 72,738,2758,16777232,1073741856,282429536508,95367431640650,
%T A253917 150094635296999148,221073919720733357899812,
%U A253917 311973482284542371301330321821976098,1329227995784915872903807060280344640,85070591730234615865843651857942052992
%N A253917 Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.
%C A253917 Intersection of A253913 and A253775.
%H A253917 Robert G. Wilson v, <a href="/A253917/b253917.txt">Table of n, a(n) for n = 1..17</a>
%H A253917 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 A253917 72 = 2^6+2+6 = 8^2+8,
%e A253917 738 = 3^6+3+6 = 9^3+9,
%e A253917 2758 = 52^2+52+2 = 14^3+14,
%e A253917 16777232 = 4^12+4+12 = 8^8+8,
%e A253917 1073741856 = 2^30+2+30 = 32^6+32,
%e A253917 282429536508 = 3^24+3+24 = 27^8+27,
%e A253917 95367431640650 = 5^20+5+20 = 25^10+25,
%e A253917 150094635296999148 = 9^18+9+18 = 27^12+27,
%e A253917 221073919720733357899812 = 6^30+6+30 = 30^15+36,
%e A253917 311973482284542371301330321821976098 = 7^42+7+42 = 49^21+49,
%e A253917 1329227995784915872903807060280344640 = 4^60+4+60 = 64^20+64,
%e A253917 85070591730234615865843651857942052992 = 2^126+2+126 = 128^18+128,
%e A253917 etc. - _Robert G. Wilson v_, Jan 19 2015
%t A253917 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 A253917 Cf. A253913, A253775, A253914, A253916.
%K A253917 nonn
%O A253917 1,1
%A A253917 _Alex Ratushnyak_, Jan 18 2015
%E A253917 a(7)-a(12) from _Robert G. Wilson v_, Jan 19 2015