login
A236507
Smallest k that is a concatenation of two numbers x and y where x^2 + y^2 = n^2.
1
10, 20, 30, 40, 34, 60, 70, 80, 90, 68, 1100, 1200, 1205, 1400, 1209, 1600, 1508, 1800, 1900, 1216, 2100, 2200, 2300, 2400, 1520, 1024, 2700, 2800, 2021, 1824, 3100, 3200, 3300, 1630, 2128, 3600, 1235, 3800, 1536, 2432, 4009, 4200, 4300, 4400, 2736, 4600, 4700
OFFSET
1,1
COMMENTS
This sequence is the union of three subsequences:
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,...
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,...
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:
26 = 2*13, 34=2*17, 51 = 3*17, 52 = 4*13, 58 = 2*29,...
LINKS
EXAMPLE
a(37) = 1235 because 1235 is the concatenation of 12 and 35, and 12^2 + 35^2 = 1369 = 37^2.
MAPLE
with(numtheory):
for n from 1 to 50 do:
ii:=0:
for k from 1 to 10^8 while(ii=0)do :
x:=convert(k, base, 10):n1:=nops(x):
if irem(n1, 2)=0
then
s:=sum('x[i]*10^(i-1) ', 'i'=1..n1/2):
s1:=sum('x[j]*10^(j-n1/2-1) ', 'j'=n1/2+1..n1):
if s^2+s1^2 = n^2
then
ii:=1: printf(`%d, `, k):
else
fi:
fi:
od:
od:
CROSSREFS
Sequence in context: A300024 A069534 A237416 * A031140 A095973 A299970
KEYWORD
nonn,base,less
AUTHOR
Michel Lagneau, Jan 27 2014
STATUS
approved