login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A275986 Positive integers of the form x*10^k + y which also equal x^2 + y^2 (x, y and k being positive integers). 0
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 (list; graph; refs; listen; history; text; internal format)
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: A290827 A290835 A290549 * A215119 A210169 A027900

Adjacent sequences:  A275983 A275984 A275985 * A275987 A275988 A275989

KEYWORD

nonn

AUTHOR

Douglas E. Iannucci, Aug 15 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 15 22:19 EDT 2019. Contains 324145 sequences. (Running on oeis4.)