A062892 -id:A062892 - cp's OEIS frontend

A096599 Squares k^2 with property that A062892(k^2) = 1.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 225, 289, 324, 361, 484, 529, 576, 676, 729, 784, 841, 1156, 1225, 1444, 1521, 1681, 1849, 2116, 2209, 2601, 2704, 3025, 3136, 3249, 3364, 3481, 3721, 3844, 3969, 4225, 4356, 4489, 4624, 5041, 5184, 5329, 5476

Offset: 1

Reinhard Zumkeller, Jun 29 2004

David A. Corneth, Table of n, a(n) for n = 1..6883

Cf. A096598, A096600, A000290, A000037, A039986.

from math import isqrt
from sympy.utilities.iterables import multiset_permutations as mp
def sqr(n): return isqrt(n)**2 == n
def ok(square):
    s = str(square)
    perms = (int("".join(p)) for p in mp(s, len(s)))
    return len(set(p for p in perms if sqr(p))) == 1
def aupto(limit): return [k*k for k in range(isqrt(limit)+1) if ok(k*k)]
print(aupto(5476)) # Michael S. Branicky, Oct 18 2021

Definition clarified by N. J. A. Sloane, Jan 16 2014

A068805 Suppose the integer m has k decimal digits; make a list of the k! strings obtained by permuting the digits in all possible ways; discard any leading zeros; count distinct squares in the list (A062892); a(n) = smallest m that yields n squares.

Original entry on oeis.org

1, 100, 169, 10269, 13468, 10044, 100269, 1000269, 10069, 100069, 1001466, 1000044, 10012689, 10045669, 10001466, 1003468, 10023469, 1000069, 10000069, 10002456, 10003468, 100045669, 100023469, 100001466, 100124469, 100045678, 100345689, 100023489, 100000069, 100002456

Offset: 1

Amarnath Murthy, Mar 06 2002

			a(3) = 169 whose 3 permutations 169, 196 and 961 yield three different squares.

David A. Corneth, Table of n, a(n) for n = 1..1968

Cf. A062892, A046891, A046892.

a=Table[0, {15}]; Do[b=Count[ IntegerQ /@ Sqrt[ FromDigits /@ Permutations[ IntegerDigits[n]]], True]; If[b<15&&a[[b]]==0, a[[b]]=n], {n, 1, 287618} ] (* Robert G. Wilson v, May 22 2003 *)

More terms from Robert G. Wilson v, May 22 2003
a(13)-a(20) from John W. Layman, Sep 27 2004
More terms from David A. Corneth, Oct 18 2021

A007937 Nonsquares such that some permutation of digits is a square.

Original entry on oeis.org

10, 18, 40, 46, 52, 61, 63, 90, 94, 106, 108, 112, 136, 148, 160, 163, 180, 184, 205, 211, 234, 243, 250, 252, 259, 265, 279, 295, 297, 298, 306, 316, 342, 360, 406, 409, 414, 418, 423, 432, 448, 460, 478, 481, 487, 490, 502, 520, 522, 526, 562, 567, 592

Offset: 1

R. Muller

Harvey P. Dale, Table of n, a(n) for n = 1..2500
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.

Cf. A000290, A062892, A096600.

Select[Range[600],!IntegerQ[Sqrt[#]]&&AnyTrue[FromDigits/@ Permutations[ IntegerDigits[ #]],IntegerQ[ Sqrt[#]]&]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 17 2020 *)

from math import isqrt
from sympy.utilities.iterables import multiset_permutations as mp
def sqr(n): return isqrt(n)**2 == n
def ok(n):
    if sqr(n): return False
    s = str(n)
    return any(sqr(int("".join(p))) for p in mp(s, len(s)))
print([k for k in range(600) if ok(k)]) # Michael S. Branicky, Oct 18 2021

A062892(a(n)) > 0.

More terms from Reinhard Zumkeller, Jun 29 2004

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.

cp's OEIS Frontend

A096599 Squares k^2 with property that A062892(k^2) = 1.

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A068805 Suppose the integer m has k decimal digits; make a list of the k! strings obtained by permuting the digits in all possible ways; discard any leading zeros; count distinct squares in the list (A062892); a(n) = smallest m that yields n squares.

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A007937 Nonsquares such that some permutation of digits is a square.

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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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