A275986 Positive integers of the form x*10^k + y which also equal x^2 + y^2 (x, y and k being positive integers).

101, 1233, 8833, 10001, 10100, 990100, 1000001, 5882353, 94122353, 99009901, 100000001, 100010000, 1765038125, 2584043776, 7416043776, 8235038125, 9901009901, 10000000001, 48600220401, 116788321168, 123288328768, 601300773101, 876712328768, 883212321168, 990100990100, 999900010000, 1000000000001, 1000001000000

Offset

1,1

Comments

The condition x^2 + y^2 = x*10^k + y is equivalent to (2x-10^k)^2 + (2y-1)^2 = 10^2k + 1, so to find these sequence elements it is necessary to write 10^2k + 1 as the sum of two squares.

The number of elements in this sequence corresponding to a fixed k is tau(10^2k + 1) - 1, where tau counts the (positive) divisors of a natural number. For all k, 10^2k + 1 is itself a member of the sequence corresponding to k, and is the only one such if it is prime. The elements themselves are arranged according to magnitude, indexed here by n. There is some disruption of the order of the terms versus the corresponding exponent k. For example, the twelfth member of the sequence, 100010000, corresponds to k=6, yet the thirteenth, 1765038125, corresponds to the smaller k=5.

Contains 10^(2*i) + 10^(4*i) and 10^(6*i) - 10^(4*i) + 10^(2*i) for each i >= 1 (corresponding to k = 3*i). - Robert Israel, Mar 30 2017

Links

Table of n, a(n) for n=1..28.

Steven Charlton, Square sum concatenation - Number theory challenge

A. van der Poorten, K. Thomsen, and M. Wiebe, A curious cubic identity and self-similar sums of squares, The Mathematical Intelligencer, v.29(2), pp. 69-73, June 2007.

Example

a(1) = 101 corresponds to k = 1, x = 10, and y = 1.
a(2) = 1233 corresponds to k = 2, x = 12, y = 33.

Crossrefs

Sequence in context: A290835 A290549 A354605 * A368416 A215119 A210169

Adjacent sequences: A275983 A275984 A275985 * A275987 A275988 A275989

Keyword

nonn

Author

Douglas E. Iannucci, Aug 15 2016

Status

approved