A377733 Numbers k such that k and k+1 are both terms in A377732.

3, 63, 154, 155, 323, 579, 583, 903, 978, 1023, 2019, 2499, 3503, 5174, 5183, 5379, 8234, 9603, 11534, 12415, 14718, 16383, 20454, 20538, 26243, 31930, 39999, 46814, 58563, 69719, 82943, 90218, 93995, 96663, 102943, 114243, 117998, 118979, 124118, 135814, 138490, 149879

Offset: 1

Author

Amiram Eldar, Nov 05 2024

Keywords

• nonn

Comments

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).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5242 (terms below 10^10)
• Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, Integers with a sum of co-divisors yielding a square, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; author's copy.

Crossrefs

Subsequence of A377732.

A377736 is a subsequence.

Cf. A141046.

Programs

Mathematica

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] &]

PARI

is1(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}