A088959 Lowest numbers which are d-Pythagorean decomposable, i.e., square is expressible as sum of two distinct positive squares in more ways than for any smaller number.

1, 5, 25, 65, 325, 1105, 5525, 27625, 32045, 160225, 801125, 1185665, 5928325, 29641625, 48612265, 243061325, 1215306625, 2576450045, 12882250225, 64411251125, 157163452745, 785817263725, 3929086318625, 10215624428425, 11472932050385, 51078122142125

Offset

1,2

References

R. M. Sternheimer, Additional Remarks Concerning The Pythagorean Triplets, Journal of Recreational Mathematics, Vol. 30, No. 1, pp. 45-48, 1999-2000, Baywood NY.

Links

Ray Chandler, Table of n, a(n) for n = 1..307

Example

From Petros Hadjicostas, Jul 21 2019: (Start)
Squares 1^2, 2^2, 3^2, and 4^2 have 0 representations as the sum of two distinct positive squares. (Thus, A088111(1) = 0 for the number of representations of 1^2.) Thus a(1) = 1.
Square 5^2 can be written as 3^2 + 4^2 only (here A088111(2) = 1). Thus, a(2) = 5.
Looking at sequence A046080, we see that for 5 <= n <= 24, only n^2 = 5^2, 10^2, 13^2, 15^2, 17^2, 20^2 can be written as a sum of two distinct positive squares (in a single way) because 5^2 = 3^2 + 4^2, 10^2 = 6^2 + 8^2, 13^2 = 5^2 + 12^2, 17^2 = 8^2 + 15^2, and 20^2 = 12^2 + 16^2.
Since A046080(25) = 2 and A088111(3) = 2, we have that 25^2 can be written as a sum of two distinct positive squares in two ways. Indeed, 25^2 = 7^2 + 24^2 = 15^2 + 20^2. Thus, a(3) = 25.
For 26 <= n <= 64, we see from sequence A046080 that n^2 cannot be written in more than 2 ways as a sum of two distinct positive squares.
Since A046080(65) = 4, we see that 65^2 can be written as the sum of two distinct positive squares in 4 ways. Indeed, 65^2 = 16^2 + 63^2 = 25^2 + 60^2 = 33^2 + 56^2 = 39^2 + 52^2. Thus, a(4) = 65.
(End)

Crossrefs

Cf. A052199. Subsequence of A054994. Number of ways: see A088111. Where records occur in A046080.

Sequence in context: A108403 A007058 A071383 * A018782 A340388 A146665

Adjacent sequences: A088956 A088957 A088958 * A088960 A088961 A088962

Keyword

nonn

Author

Lekraj Beedassy, Dec 01 2003

Extensions

Corrected and extended by Ray Chandler, Jan 12 2012

Name edited by Petros Hadjicostas, Jul 21 2019

Status

approved