A062892 Number of squares that can be obtained by permuting the digits of n.

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

Author

Amarnath Murthy, Jun 29 2001

Keywords

• base
• nonn
• easy

Comments

The original definition was ambiguous (it did not specify how repeated digits or leading zeros are to be handled). Here is a precise version (based on the Mathematica code):

Suppose the decimal expansion of n has d digits. Apply all d! permutations, discard duplicates, but keep any with leading zeros; now ignore leading zeros; a(n) is the number of squares on the resulting list. For example, if n = 100 we end up with 100, 010, 001, and both 100 and 1 are squares, so a(100)=2. If n=108 we get 6 numbers but only (0)81 is a square, so a(108)=1. - N. J. A. Sloane, Jan 16 2014

Examples

			a(169) = 3; the squares obtained by permuting the digits are 169, 196, 961.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999

Crossrefs

A096599 gives the squares k^2 such that a(k^2) = 1.

Programs

Mathematica

Table[t1=Table[FromDigits[k],{k,Permutations[IntegerDigits[n]]}]; p=Length[Select[t1,IntegerQ[Sqrt[#]]&]], {n,0,104}] (* Jayanta Basu, May 17 2013 *)

PARI

a(n) = {my(d = vecsort(digits(n)), res = 0); forperm(d, p, res += issquare(fromdigits(Vec(p)))); res } \\ David A. Corneth, Oct 18 2021

Python

from math import isqrt
from sympy.utilities.iterables import multiset_permutations as mp
def sqr(n): return isqrt(n)**2 == n
def a(n):
    s = str(n)
    perms = (int("".join(p)) for p in mp(s, len(s)))
    return len(set(p for p in perms if sqr(p)))
print([a(n) for n in range(105)] ) # Michael S. Branicky, Oct 18 2021

Formula

a(A096600(n))=0; a(A007937(n))>0; a(A096599(n))=1; a(A096598(n))>1. - Reinhard Zumkeller, Jun 29 2004

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 02 2001