A211996 -id:A211996 - cp's OEIS frontend

A218381 Numbers k such that A211996(k) is not zero.

3, 4, 8, 14, 15, 18, 20, 24, 28, 35, 39, 46, 48, 55, 60, 63, 64, 66, 68, 80, 84, 94, 99, 100, 114, 120, 124, 126, 128, 136, 138, 143, 144, 150, 154, 155, 156, 158, 168, 180, 183, 195, 196, 203, 220, 224, 234, 238, 240, 243, 255, 258, 260, 275, 284, 288, 291

Offset: 1

Michel Marcus, Oct 27 2012

nonn

For any n, the equation x^4 + a(n)*y^4 = z^2 is solvable in integers. - Arkadiusz Wesolowski, Aug 15 2013

The asymptotic density of this sequence is 0 (De Koninck et al., 2024). - Amiram Eldar, Nov 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
David Clark, An arithmetical function associated with the rank of elliptic curves, Canad. Math. Bull. Vol. 34 (2), (1991), pp. 181-185.
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. A211996, A218382, A228880.

q[k_] := DivisorSum[k, 1 &, #^2 <= k && IntegerQ[Sqrt[# + k/#]] &] > 0; Select[Range[300], q] (* Amiram Eldar, Nov 05 2024 *)

is(k) = k > 1 && fordiv(k, d, if(issquare(d + k/d), return(1)); if(d^2 > k, return(0))); \\ Amiram Eldar, Nov 05 2024

A218382 Indices of records in A211996.

Original entry on oeis.org

1, 3, 48, 828, 8820, 26100, 417600, 2114100, 3009600, 43864128, 200278848, 1013911668, 7823392500, 16222586688, 259561387008, 633694792500, 4152982192128

Offset: 1

Michel Marcus, Oct 27 2012

a(18) <= 10139116680000. - Donovan Johnson, Nov 24 2012

D. Clark, An arithmetical function associated with the rank of elliptic curves, Canad. Math. Bull. Vol. 34 (2), 1991 pp. 181-185.

Cf. A211996, A218381.

a(12)-a(17) from Donovan Johnson, Nov 24 2012

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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A218381 Numbers k such that A211996(k) is not zero.

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A218382 Indices of records in A211996.

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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

A218381 Numbers k such that A211996(k) is not zero.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A218382 Indices of records in A211996.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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