

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
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OFFSET

1,1


COMMENTS

The condition x^2 + y^2 = x*10^k + y is equivalent to (2x10^k)^2 + (2y1)^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 selfsimilar sums of squares, The Mathematical Intelligencer, v.29(2), pp. 6973, 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



