A295787 - OEIS

%I #26 Sep 07 2020 16:17:50

%S 126,127,350,351,441,485,511,848,1431,1568,2024,2752,2843,3024,3844,

%T 4697,5489,7120,7343,7399,8125,8126,8623,9430,9800,10703,10842,11474,

%U 12176,12335,12742,12743,13748,14191,14911,15254,16128,16640,16857,17067,17207,18095,18567

%N Positive integers m such that m, m + 1 and m + 2 are a sum of a positive square and a positive cube.

%C Is a(n) >= c*n^e for some constants c and e? For terms in the b-file, we'd have e > 2.1598. - _David A. Corneth_, Mar 15 2019

%H David A. Corneth, <a href="/A295787/b295787.txt">Table of n, a(n) for n = 1..14685</a> (Terms <= 10^9)

%H Bundeswettbewerb Mathematik 2017, <a href="https://www.mathe-wettbewerbe.de/bwm/aufgaben/aufgaben-2017/aufgaben-17-2.pdf">Die Aufgaben der 2. Runde 2017</a>

%H Bundeswettbewerb Mathematik 2017, <a href="https://www.mathe-wettbewerbe.de/bwm/aufgaben/aufgaben-2017/loes-17-2-e.pdf">Aufgaben und Lösungen 2. Runde 2017</a>

%H David A. Corneth, <a href="/A295787/a295787.gp.txt">PARI program</a>

%e 126 and 127 are terms because: 126 = 1^2 + 5^3, 127 = 10^2 + 3^3, 128 = 8^2 + 4^3, 129 = 11^2 + 2^3. - _Bernard Schott_, Mar 17 2019

%t s = Union@ Flatten@ Table[s^2 + c^3, {s, 141}, {c, 27}]; First@# & /@ Select[Partition[s, 3, 1], #[[1]] + 2 == #[[3]] &] (* _Robert G. Wilson v_, Jan 07 2018 *)

%t With[{mx=19000},Select[Partition[Union[Flatten[Table[a^2+b^3,{a,Ceiling[ Sqrt[mx]]},{b,Ceiling[Surd[mx,3]]}]]],3,1],Differences[#]=={1,1}&]][[All,1]] (* _Harvey P. Dale_, Sep 07 2020 *)

%o (PARI) is_a055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0 \\ after _Charles R Greathouse IV_

%o is(n) = is_a055394(n) && is_a055394(n+1) && is_a055394(n+2) \\ _Felix Fröhlich_, Jan 08 2018

%o (PARI) See Corneth Link \\ _David A. Corneth_, Mar 15 2019

%Y Cf. A055394.

%K nonn

%O 1,1

%A _David A. Corneth_, Dec 30 2017