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 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 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

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