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

0,101

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A361162 A118626 A277149 * A118553 A102448 A102683

Adjacent sequences: A062889 A062890 A062891 * A062893 A062894 A062895

Keyword

base,nonn,easy

Author

Amarnath Murthy, Jun 29 2001

Extensions

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

Status

approved