A045877 -id:A045877 - cp's OEIS frontend

A035126 Squares when digits rotated right once remain square.

1, 4, 9, 256, 441, 961, 16641, 48841, 61009, 66564, 1127844, 2537649, 3857296, 4932841, 182682256, 298840369, 342842256, 392872041, 493772841, 787588096, 877996161, 10766967696, 33255899044, 49382172841, 74825772849

Offset: 1

Patrick De Geest, Nov 15 1998

Those resulting in leading zeros excluded.

			2221^2 = 4932841 -> 1493284 = 1222^2. Note that the root behaves accordingly!

Chai Wah Wu, Table of n, a(n) for n = 1..1838
Eric Weisstein's World of Mathematics, Square Number

Cf. A045877, A035130.

[k:k in [m^2:m in [1..10^6]]| IsSquare(Seqint( (Intseq(Floor(k/10)) cat  [ Intseq(k)[1]])))]; // Marius A. Burtea, Oct 08 2019

Select[Range[300000]^2,IntegerQ[Sqrt[FromDigits[RotateRight[ IntegerDigits[ #]]]]]&] (* Harvey P. Dale, Mar 22 2015 *)

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A035126_gen(): # generator of terms
    for l in count(0):
        l1, l2 = 10**(l+1), 10**l
        yield from sorted(set(x**2 for z in (diop_DN(10,m*(1-l1)) for m in range(10)) for x, y in z if l1 >= x**2 >= l2))
A035126_list = list(islice(A035126_gen(),30)) # Chai Wah Wu, Apr 23 2022

a(n) = A045877(n)^2. - R. J. Mathar, Jan 25 2017

A045878 Numbers k such that rotating digits of k^2 left once still yields a square.

Original entry on oeis.org

1, 2, 3, 12, 14, 25, 108, 122, 216, 310, 1222, 2028, 2527, 3042, 11802, 12222, 13704, 24865, 25185, 26053, 30494, 122222, 208148, 247137, 312222, 1125786, 1222222, 1325080, 2084388, 2551071, 3025794, 3037736, 3126582, 10716846, 10787208

Offset: 1

Erich Friedman

Squares resulting in leading zeros are excluded.

A090843 is a subsequence. - Chai Wah Wu, Apr 23 2022

			11303148^2 = {1}27761154709904 -> 277611547099041{1} = 16661679^2.

Eric Weisstein's World of Mathematics, Square Number

Cf. A000290, A045877, A035129, A090843.

rlsQ[n_]:=Module[{idnrl=RotateLeft[IntegerDigits[n^2]]},First[idnrl]>0 && IntegerQ[Sqrt[FromDigits[idnrl]]]]; Select[Range[11000000],rlsQ] (* Harvey P. Dale, Nov 03 2013 *)

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A045878_gen(): # generator of terms
    for l in count(0):
        l1, l2 = 10**(l+1), 10**l
        yield from sorted(set(abs(y) for z in (diop_DN(10,m*(1-l1)) for m in range(10)) for x, y in z if l1 >= x**2 >= l2))
A045878_list = list(islice(A045878_gen(), 30)) # Chai Wah Wu, Apr 23 2022

More terms from Patrick De Geest, Nov 15 1998

A035128 Rotating digits of a(n)^3 right once still yields a cube.

Original entry on oeis.org

1, 2, 5, 379

Offset: 1

Patrick De Geest, Nov 15 1998

Cubes resulting in leading zeros excluded.

			379^3 = 54439939 -> 95443993 = 457^3.

Eric Weisstein's World of Mathematics, Cubic Number

Cf. A045877, A035129, A035130.

[k:k in [1..100000]| IsPower(Seqint((Intseq(Floor(k^3/10)) cat [Intseq(k^3)[1]])),3)]; // Marius A. Burtea, Oct 08 2019

Select[Range[500], IntegerQ @ Surd[FromDigits @ RotateRight @ IntegerDigits[#^3], 3] &] (* Amiram Eldar, Oct 08 2019 *)

cp's OEIS Frontend

A035126 Squares when digits rotated right once remain square.

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A045878 Numbers k such that rotating digits of k^2 left once still yields a square.

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A035128 Rotating digits of a(n)^3 right once still yields a cube.

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Examples

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Programs

Magma

Mathematica