cp's OEIS Frontend

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

%I A275986 #23 Mar 31 2017 03:47:52
%S A275986 101,1233,8833,10001,10100,990100,1000001,5882353,94122353,99009901,
%T A275986 100000001,100010000,1765038125,2584043776,7416043776,8235038125,
%U A275986 9901009901,10000000001,48600220401,116788321168,123288328768,601300773101,876712328768,883212321168,990100990100,999900010000,1000000000001,1000001000000
%N A275986 Positive integers of the form x*10^k + y which also equal x^2 + y^2 (x, y and k being positive integers).
%C A275986 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.
%C A275986 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.
%C A275986 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
%H A275986 Steven Charlton, <a href="http://community.dur.ac.uk/steven.charlton/squaresumcat">Square sum concatenation - Number theory challenge</a>
%H A275986 A. van der Poorten, K. Thomsen, and M. Wiebe, <a href="http://dx.doi.org/10.1007/BF02986204">A curious cubic identity and self-similar sums of squares</a>, The Mathematical Intelligencer, v.29(2), pp. 69-73, June 2007.
%e A275986 a(1) = 101 corresponds to k = 1, x = 10, and y = 1.
%e A275986 a(2) = 1233 corresponds to k = 2, x = 12, y = 33.
%K A275986 nonn
%O A275986 1,1
%A A275986 _Douglas E. Iannucci_, Aug 15 2016