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A365295 a(n) is the least positive integer that can be expressed as the sum of two distinct perfect powers (A001597) in exactly n ways.

%I A365295 #42 Mar 03 2024 09:29:54
%S A365295 1,5,17,129,468,1025,2628,12025,32045,27625,138125,430625,204425,
%T A365295 160225,2010025,2348125,801125,1743625,2082925,4978025,4005625,
%U A365295 12325625,30525625,73046025,5928325,13287625,46437625,45177925,35409725,120737825,52073125,66438125,29641625,32846125,956974625
%N A365295 a(n) is the least positive integer that can be expressed as the sum of two distinct perfect powers (A001597) in exactly n ways.
%H A365295 Karl-Heinz Hofmann and Hugo Pfoertner, <a href="/A365295/b365295.txt">Table of n, a(n) for n = 0..77</a>
%H A365295 Karl-Heinz Hofmann, <a href="/A365295/a365295_1.txt">Python program</a>
%H A365295 Hugo Pfoertner, <a href="/A365295/a365295.txt">PARI program and results (terms < 2*10^9)</a>, Sep 10 2023.
%e A365295 For n = 2: a(2) = 17 = 1^2 + 2^4 = 2^3 + 3^2.
%e A365295 a(6) = 2628 via 3^3 + 51^2 = 2^7 + 50^2 = 18^2 + 48^2 = 21^2 + 3^7 = 2^9 + 46^2 = 30^2 + 12^3. - _David A. Corneth_, Sep 09 2023
%o A365295 (PARI) upto(n) = {n = (sqrtint(n) + 1)^2; my(v = vector(n), pows = List([1]), r = -1, res = []); for(j = 2, logint(n, 2), for(i = 2, sqrtnint(n, j), listput(pows, i^j))); pows = Set(pows); for(i = 1, #pows - 1, j = i+1; c = pows[i] + pows[j]; while(c <= n, v[c]++; j++; c = pows[i] + pows[j])); for(i = 1, #v, c = v[i]+1; if(c > #res, res = concat(res, vector(c - #res, j, oo))); if(i < res[c], res[c] = i)); res} \\ _David A. Corneth_, Sep 08 2023
%o A365295 (PARI) \\ see link
%o A365295 (Python) # see link
%Y A365295 Cf. A001597, A093195, A362424, A363040.
%K A365295 nonn
%O A365295 0,2
%A A365295 _Ilya Gutkovskiy_, Aug 31 2023
%E A365295 a(8)-a(10) from _David Consiglio, Jr._, Sep 08 2023
%E A365295 a(9) corrected and a(11)-a(34) from _Hugo Pfoertner_, Sep 10 2023