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%I #6 Jan 27 2014 17:56:44
%S 10,20,30,40,34,60,70,80,90,68,1100,1200,1205,1400,1209,1600,1508,
%T 1800,1900,1216,2100,2200,2300,2400,1520,1024,2700,2800,2021,1824,
%U 3100,3200,3300,1630,2128,3600,1235,3800,1536,2432,4009,4200,4300,4400,2736,4600,4700
%N Smallest k that is a concatenation of two numbers x and y where x^2 + y^2 = n^2.
%C This sequence is the union of three subsequences:
%C a subsequence of trivial numbers with n different from 5k, k=1,2,... and y = 0. This subsequence is 10, 20, 30, 40, 60, 70, 80, 90, 1100, 1200, 1400, 1600, 1800, 1900, 2100,...
%C a subsequence with n = 5k and y different from 0. This subsequence is 34, 68, 1209, 1216, 1520, 1824, 2128, 2432, 2736, 1448, 3344, 3648, 1663, 4256, 2172, 4864, 1384,...
%C a subsequence with n different from 5k and y different from 0. This subsequence is 1205, 1508, 1024, 2021, 1630, 1235, 1636, 4009,... for n = 13, 17, 26, 29, 34, 37, 39, 41,... Except for the prime 5, the prime indices of this subsequence are Pythagorean primes 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173,... (A002144) and the composite indices are multiples of Pythagorean primes:
%C 26 = 2*13, 34=2*17, 51 = 3*17, 52 = 4*13, 58 = 2*29,...
%H Michel Lagneau, <a href="/A236507/b236507.txt">Table of n, a(n) for n = 1..500</a>
%e a(37) = 1235 because 1235 is the concatenation of 12 and 35, and 12^2 + 35^2 = 1369 = 37^2.
%p with(numtheory):
%p for n from 1 to 50 do:
%p ii:=0:
%p for k from 1 to 10^8 while(ii=0)do :
%p x:=convert(k,base,10):n1:=nops(x):
%p if irem(n1,2)=0
%p then
%p s:=sum('x[i]*10^(i-1) ', 'i'=1..n1/2):
%p s1:=sum('x[j]*10^(j-n1/2-1) ', 'j'=n1/2+1..n1):
%p if s^2+s1^2 = n^2
%p then
%p ii:=1: printf(`%d, `,k):
%p else
%p fi:
%p fi:
%p od:
%p od:
%Y Cf. A002144, A236383.
%K nonn,base,less
%O 1,1
%A _Michel Lagneau_, Jan 27 2014
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