A048653 Numbers k such that the decimal digits of k^2 can be partitioned into two or more nonzero squares.

7, 12, 13, 19, 21, 35, 37, 38, 41, 44, 57, 65, 70, 107, 108, 112, 119, 120, 121, 125, 129, 130, 190, 191, 204, 205, 209, 210, 212, 223, 253, 285, 305, 306, 315, 342, 343, 345, 350, 369, 370, 379, 380, 408, 410, 413, 440, 441, 475, 487, 501, 538, 570, 642, 650

Offset

1,1

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Example

12 is present because 12^2=144 can be partitioned into three squares 1, 4 and 4.
108^2 = 11664 = 1_16_64, 120^2 = 14400 = 1_4_400, so 108 and 120 are in the sequence.

Mathematica

(* This non-optimized program is not suitable to compute a large number of terms. *) split[digits_, pos_] := Module[{pos2}, pos2 = Transpose[{Join[ {1}, Most[pos+1]], pos}]; FromDigits[Take[digits, {#[[1]], #[[2]]}]]& /@ pos2]; sel[n_] := Module[{digits, ip, ip2, accu, nn}, digits = IntegerDigits[n^2]; ip = IntegerPartitions[Length[digits]]; ip2 = Flatten[ Permutations /@ ip, 1]; accu = Accumulate /@ ip2; nn = split[ digits, #]& /@ accu; SelectFirst[nn, Length[#]>1 && Flatten[ IntegerDigits[#] ] == digits && AllTrue[#, #>0 && IntegerQ[Sqrt[#]]&]&] ]; k = 1; Reap[Do[If[(s = sel[n]) != {}, Print["a(", k++, ") = ", n, " ", n^2, " ", s]; Sow[n]], {n, 1, 10^4}]][[2, 1]] (* Jean-François Alcover, Sep 28 2016 *)

Prog

(Haskell)
a048653 n = a048653_list !! (n-1)
a048653_list = filter (f . show . (^ 2)) [1..] where
   f zs = g (init $ tail $ inits zs) (tail $ init $ tails zs)
   g (xs:xss) (ys:yss)
     | h xs      = h ys || f ys || g xss yss
     | otherwise = g xss yss
     where h ds = head ds /= '0' && a010052 (read ds) == 1
   g _ _ = False
-- Reinhard Zumkeller, Oct 11 2011
(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 d[i] != '0' and issquare(int(d[:i])) and ok(int(d[i:]), c+1):
            return True
    return False
def aupto(lim): return [r for r in range(lim+1) if ok(r*r, 1)]
print(aupto(650)) # Michael S. Branicky, Jul 10 2021

Crossrefs

Cf. A048646, A048375, A010052, A000290; subsequence of A128783.

Sequence in context: A328414 A083681 A178660 * A205807 A075696 A061120

Adjacent sequences: A048650 A048651 A048652 * A048654 A048655 A048656

Keyword

base,nice,nonn

Author

Felice Russo

Extensions

Corrected and extended by Naohiro Nomoto, Sep 01 2001

Definition clarified by Harvey P. Dale, May 09 2021

Status

approved