cp's OEIS Frontend

A178530 Numbers k with the property that there exist nonnegative integers a and b such that k = concat(a,b) = a^2+b^2.

%I A178530 #28 Jul 14 2025 13:08:02
%S A178530 0,1,100,101,1233,8833,10100,990100,5882353,94122353,1765038125,
%T A178530 2584043776,7416043776,8235038125,116788321168,123288328768,
%U A178530 876712328768,883212321168,7681802663025,8896802846976,13793103448276,15348303604525,84651703604525,86206903448276,91103202846976,92318202663025,106058810243728
%N A178530 Numbers k with the property that there exist nonnegative integers a and b such that k = concat(a,b) = a^2+b^2.
%C A178530 The sum of two numbers a1 and a2 that share a common b has the form of 10^j. Example: 12 + 88 = 100
%C A178530 The ordered pair of the final digit of a and b is always one of (0,0), (0,1), (0,5), (0,6), (2,3), (8,3), (2,8), or (8,8).
%C A178530 If b has k decimal digits, then (2a - 10^k)^2 + (2b - 1)^2 = 10^(2k) + 1 giving a way for efficient computation of many terms. - _Max Alekseyev_, Aug 17 2013
%H A178530 Max Alekseyev, <a href="/A178530/b178530.txt">Table of n, a(n) for n = 1..200</a>
%e A178530 0 = 0^2+0^2 [this seems a bit far-fetched. - _N. J. A. Sloane_, Dec 23 2010]
%e A178530 1=0^2+1^2 [ditto]
%e A178530 100=10^2+0^2.
%e A178530 101=10^2+1^2.
%e A178530 1233=12^2+33^2.
%t A178530 Sort[Reap[Do[n=a^2+b^2; If[n==FromDigits[Join[IntegerDigits[a], IntegerDigits[b]]], Sow[n]], {a,0, 1000}, {b, 0, 1000}]][[2, 1]]]
%Y A178530 See A055616, A064942, A101311 for closely related sequences.
%K A178530 nonn,base
%O A178530 1,3
%A A178530 _Karsten Meyer_, Dec 23 2010
%E A178530 Edited by _N. J. A. Sloane_, Dec 23 2010
%E A178530 a(11)-a(14) from _Nathaniel Johnston_, Jan 03 2011
%E A178530 Terms a(15) onward from _Max Alekseyev_, Aug 17 2013