login
A244283
Consider a number n with m decimal digits, m>9. The sequence lists the numbers n such that the prefix of length m-1 and the suffix of length m-1 are both perfect squares.
3
10, 11, 14, 19, 40, 41, 44, 49, 90, 91, 94, 99, 164, 364, 649, 816, 1000, 1001, 1004, 1009, 1441, 1961, 2256, 4000, 4001, 4004, 4009, 4841, 6256, 7841, 9000, 9001, 9004, 9009, 20256, 30256, 31369, 40961, 46241, 51849, 54761, 60841, 73969, 79216, 90256, 94096
OFFSET
1,1
COMMENTS
Let x(0)x(1)... x(q-1)x(q) denote the decimal expansion of a number n. The sequence lists the numbers n such that the prefix x(0)x(1)... x(q-1) and the suffix x(1)... x(q-1)x(q) are both a perfect square.
The primes of the sequence are 11, 19, 41, 1009, 4001, 7841, 9001, 40961,...
EXAMPLE
816 is in the sequence because 81 and 16 are squares.
MAPLE
with(numtheory):
for n from 10 to 20000 do:
x:=convert(n, base, 10):n1:=nops(x):
s1:=sum('x[i]*10^(i-1) ', 'i'=1..n1-1):
s2:=(n-irem(n, 10))/10:ss1:=sqrt(s1):ss2:=sqrt(s2):
if ss1=floor(ss1) and ss2=floor(ss2)
then
printf(`%d, `, n):
else
fi:
od:
PROG
(PARI) isok(n) = (left = n\10) && issquare(left) && (pt = 10^(#Str(n)-1)) && issquare(n - (n\pt)*pt); \\ Michel Marcus, Jun 25 2014
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Michel Lagneau, Jun 25 2014
STATUS
approved