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.

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,...

Links

Table of n, a(n) for n=1..46.

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

Cf. A000290, A046030, A244282.

Sequence in context: A062844 A144980 A261713 * A043583 A043773 A043779

Adjacent sequences: A244280 A244281 A244282 * A244284 A244285 A244286

Keyword

nonn,base

Author

Michel Lagneau, Jun 25 2014

Status

approved