A096599 -id:A096599 - cp's OEIS frontend

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

Amarnath Murthy, Jun 29 2001

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

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

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

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

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

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

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

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

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

A096600 Numbers such that in decimal representation all permutations of digits are nonsquares.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87

Offset: 1

Reinhard Zumkeller, Jun 29 2004

A062892(a(n)) = 0.

			134=2*67, 143=11*13, 314=2*157, 341=11*31, 413=7*59 and 431=A000040(83), therefore 134, 143, 314, 341, 413 and 431 are terms.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A007937, A096599, A000037, A067013.

Select[Range[100],NoneTrue[Sqrt[#]&/@(FromDigits/@Permutations[IntegerDigits[ #]]),IntegerQ]&] (* Harvey P. Dale, Dec 04 2022 *)

A096598 Squares such that some permutation of digits is also a square (in decimal representation).

Original entry on oeis.org

100, 144, 169, 196, 256, 400, 441, 625, 900, 961, 1024, 1089, 1296, 1369, 1600, 1764, 1936, 2025, 2304, 2401, 2500, 2809, 2916, 3600, 4096, 4761, 4900, 6400, 7056, 8100, 9025, 9216, 9604, 9801, 10000, 10201, 10404, 10609, 10816, 11025

Offset: 1

Reinhard Zumkeller, Jun 29 2004

A062892(a(n)) > 1.

			1024 = 32^2 and also 2401=49^2, therefore 1024 (and 2401) is a term.

Cf. A096599, A007937, A000290, A055387.

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

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A096600 Numbers such that in decimal representation all permutations of digits are nonsquares.

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Mathematica

A096598 Squares such that some permutation of digits is also a square (in decimal representation).

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