A019547 Squares which are a decimal concatenation of two or more squares.

49, 100, 144, 169, 361, 400, 441, 900, 1225, 1369, 1444, 1600, 1681, 1936, 2500, 3249, 3600, 4225, 4900, 6400, 8100, 9025, 9409, 10000, 10404, 11025, 11449, 11664, 12100, 12544, 14161, 14400, 14641, 15625, 16641, 16900, 19044, 19600, 22500, 25600, 28900

Offset: 1

Author

R. Muller

Keywords

• nonn
• base

Comments

0 counts as a square here.

Examples

			1369 is a term as it can be partitioned as 1, 36 and 9. 1444 is a term as it can be partitioned as 1, 4, 4, 4. Again, 100 is 1, 0, 0.
		

References

• L. Widmer, Construction of Elements of the Smarandache Square-Partial-Digital Sequence, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 145-146.

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
• Christian N. K. Anderson, Table of n, a(n), sqrt(a(n)), all possible decompositions of a(n) into squares for n = 1..10000 (excluding solutions with leading zeros)
• Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Crossrefs

Cf. A010052, A000290, A128783.

Programs

Mathematica

r[n_, d_] := Catch@Block[{z = Length@d, t}, z < 1 || Do[If[IntegerQ@ Sqrt@ (t = FromDigits@Take[d, i]) && t < n && r[n, Take[d, i - z]], Throw@ True], {i, z}]]; Select[Range[4, 10^4]^2, r[#, IntegerDigits@ #] &] (* Giovanni Resta, Apr 30 2013 *)

Python

from math import isqrt
def issquare(n): return isqrt(n)**2 == n
def ok(n, c):
    if n%10 in { 2, 3, 7, 8}: return False
    if issquare(n) and c > 1: return True
    d = str(n)
    for i in range(1, len(d)):
        if issquare(int(d[:i])) and ok(int(d[i:]), c+1): return True
    return False
print([r*r for r in range(180) if ok(r*r, 1)]) # Michael S. Branicky, Jul 10 2021

Formula

n^2 < a(n) < 100n^2. - Charles R Greathouse IV, Sep 19 2012

a(n) = A128783(n)^2. - Michael S. Branicky, Jul 10 2021

Extensions

More terms from Christian N. K. Anderson, Apr 30 2013