A377732 -id:A377732 - cp's OEIS frontend

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

Amiram Eldar, Nov 05 2024

nonn

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).

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.

Subsequence of A377732.
A377736 is a subsequence.
Cf. A141046.

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] &]

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);}

A377736 Numbers k such that k, k+1 and k+2 are all terms in A377732.

Original entry on oeis.org

154, 282674, 3003599678

Offset: 1

Amiram Eldar, Nov 05 2024

144544673718847655 and 10931129469745989328319 are also terms (De Koninck et al., 2024, section 6).

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.

Subsequence of A377732 and A377733.

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 = is1(2), q3); for(k = 3, kmax, q3 = is1(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}

A377731 Decimal expansion of 4 * sqrt(2Pi) Gamma(5/4) / (3 * Gamma(3/4)).

Original entry on oeis.org

2, 4, 7, 2, 0, 9, 9, 5, 6, 9, 7, 3, 5, 1, 6, 2, 5, 5, 7, 9, 1, 1, 8, 0, 0, 4, 6, 2, 9, 2, 7, 0, 1, 3, 3, 9, 4, 9, 5, 6, 7, 9, 8, 4, 3, 1, 3, 6, 2, 3, 5, 5, 8, 7, 4, 1, 1, 4, 5, 7, 0, 7, 2, 6, 7, 4, 1, 3, 6, 2, 3, 6, 9, 1, 1, 7, 4, 9, 3, 3, 0, 4, 4, 7, 6, 3, 6, 2, 4, 8, 2, 2, 0, 0, 4, 9, 5, 8, 6, 3, 0, 3, 7, 0, 2

Offset: 1

Amiram Eldar, Nov 05 2024

This constant appears in asymptotic formulas related to A211996 and A377732.

			2.47209956973516255791180046292701339495679843136235...

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.

Cf. A019727, A068465, A068467, A211996, A377732.

RealDigits[4 * Sqrt[2*Pi] * Gamma[5/4] / (3 * Gamma[3/4]), 10, 120][[1]]

4 * sqrt(2*Pi) * gamma(5/4) / (3 * gamma(3/4))

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

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A377736 Numbers k such that k, k+1 and k+2 are all terms in A377732.

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A377731 Decimal expansion of 4 * sqrt(2Pi) Gamma(5/4) / (3 * Gamma(3/4)).

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cp's OEIS Frontend

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

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A377736 Numbers k such that k, k+1 and k+2 are all terms in A377732.

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Author

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Crossrefs

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PARI

A377731 Decimal expansion of 4 * sqrt(2*Pi) * Gamma(5/4) / (3 * Gamma(3/4)).

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Mathematica

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A377731 Decimal expansion of 4 * sqrt(2Pi) Gamma(5/4) / (3 * Gamma(3/4)).