A377733 - OEIS

%I #10 Nov 06 2024 04:39:57

%S 3,63,154,155,323,579,583,903,978,1023,2019,2499,3503,5174,5183,5379,

%T 8234,9603,11534,12415,14718,16383,20454,20538,26243,31930,39999,

%U 46814,58563,69719,82943,90218,93995,96663,102943,114243,117998,118979,124118,135814,138490,149879

%N Numbers k such that k and k+1 are both terms in A377732.

%C This sequence is infinite. All the positive numbers of the form A141046(m) - 1 = 4*m^4 -1 are terms (De Koninck et al., 2024, section 6).

%H Amiram Eldar, <a href="/A377733/b377733.txt">Table of n, a(n) for n = 1..5242</a> (terms below 10^10)

%H Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, <a href="https://doi.org/10.1007/s40993-024-00520-x">Integers with a sum of co-divisors yielding a square</a>, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; <a href="https://www.jeanmariedekoninck.mat.ulaval.ca/fileadmin/Documents/Publications/2023_integers_with_a_sum_of_co-divisors_yielding_a_square.pdf">author's copy</a>.

%t q[k_] := q[k] = If[IntegerQ[Sqrt[k]], IntegerQ[Sqrt[2*Sqrt[k]]], Module[{d = Divisors[k], nh}, nh = Length[d]/2; IntegerQ[Sqrt[d[[nh]] + d[[nh + 1]]]]]]; Select[Range[150000], q[#] && q[#+1] &]

%o (PARI) is1(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));

%o lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

%Y Subsequence of A377732.

%Y A377736 is a subsequence.

%Y Cf. A141046.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 05 2024