A003166 -id:A003166 - cp's OEIS frontend

A029805 Numbers k such that k^2 is palindromic in base 8.

0, 1, 2, 3, 6, 9, 11, 27, 65, 73, 79, 81, 83, 195, 219, 237, 366, 513, 543, 585, 697, 1094, 1539, 1755, 1875, 2910, 4097, 4161, 4225, 4477, 4617, 4681, 4727, 4891, 5267, 8698, 8730, 11841, 12291, 12483, 12675, 13065, 13851, 14673, 15021

Offset: 1

Patrick De Geest

The only powers of 2 in this sequence are 1 and 2. - Alonso del Arte, Feb 25 2017

			3 is in the sequence because 3^2 = 9 = 11 in base 8, which is a palindrome.
4 is not in the sequence because 4^2 = 16 = 20 in base 8, which is not a palindrome.

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A000290, A002441, A007094.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), this sequence (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

palQ[n_, b_:10] := Module[{idn = IntegerDigits[n, b]}, idn == Reverse[idn]]; Select[Range[0, 16000], palQ[#^2, 8] &] (* Harvey P. Dale, May 19 2012 *)

from itertools import count, islice
def A029805_gen(): # generator of terms
    return filter(lambda k: (s:=oct(k**2)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A029805_list = list(islice(A029805_gen(),20)) # Chai Wah Wu, Jun 23 2022

A029994 Numbers k such that k^2 is palindromic in base 9.

Original entry on oeis.org

0, 1, 2, 10, 20, 82, 91, 100, 164, 730, 820, 1460, 6562, 6643, 6724, 7300, 7381, 7462, 13124, 13642, 13660, 14281, 54050, 59050, 59860, 65620, 66430, 118100, 123010, 126286, 161410, 161896, 487750, 531442, 532171

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A007095.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), this sequence (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

pb9Q[n_]:=Module[{idn=IntegerDigits[n^2,9]},idn==Reverse[idn]]; Select[ Range[0,600000],pb9Q] (* Harvey P. Dale, Sep 29 2013 *)

A029996 Numbers k such that k^2 is palindromic in base 11.

Original entry on oeis.org

0, 1, 2, 3, 6, 12, 24, 26, 72, 84, 122, 133, 144, 244, 255, 279, 382, 732, 1332, 1464, 1596, 2414, 2664, 2796, 3062, 4476, 7992, 14642, 14763, 14884, 15984, 16105, 16226, 17326, 29284, 29405, 30626, 33675, 34701, 63546, 87246, 87852, 88578

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), this sequence (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

A029737 Numbers whose square is palindromic in base 12.

Original entry on oeis.org

0, 1, 2, 3, 13, 26, 145, 157, 169, 179, 181, 290, 292, 302, 611, 1729, 1745, 1783, 1885, 2041, 3458, 3614, 3796, 20737, 20881, 21025, 21169, 22477, 22621, 22765, 24073, 24217, 24361, 24599, 25523, 25579, 28613, 41474, 41618, 41908, 43214

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), this sequence (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

pal12Q[n_]:=Module[{idn12=IntegerDigits[n^2,12]},idn12==Reverse[idn12]]
Select[Range[0,50000],pal12Q]  (* Harvey P. Dale, Feb 06 2011 *)

A029998 Numbers k such that k^2 is palindromic in base 13.

Original entry on oeis.org

0, 1, 2, 3, 14, 28, 170, 183, 196, 209, 308, 340, 353, 366, 2198, 2380, 2562, 2898, 4026, 4242, 4396, 4578, 7078, 7662, 28562, 28731, 28900, 29069, 30772, 30941, 31110, 32813, 32982, 33151, 37374, 51510, 52360, 54942, 55449, 57124, 57293

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), this sequence (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

A030072 Numbers k such that k^2 is palindromic in base 14.

Original entry on oeis.org

0, 1, 2, 3, 15, 24, 30, 47, 165, 197, 211, 225, 239, 394, 408, 422, 2190, 2445, 2745, 2955, 3165, 5490, 5700, 8565, 38417, 38613, 38809, 39005, 41175, 41371, 41567, 41763, 43737, 43933, 44129, 48159, 55962, 76834, 77030, 77226, 79592, 79788

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), this sequence (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

pal14Q[n_]:=Module[{idn14=IntegerDigits[n^2,14]},idn14==Reverse[idn14]]; Select[Range[0,80000],pal14Q] (* Harvey P. Dale, Mar 09 2012 *)

A030073 Numbers k such that k^2 is palindromic in base 15.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 12, 16, 19, 32, 39, 64, 76, 128, 144, 226, 241, 256, 271, 311, 452, 467, 478, 482, 576, 715, 904, 964, 1024, 1748, 1808, 1868, 2304, 2652, 2860, 3376, 3401, 3616, 3856, 4639, 6752, 6992, 7172, 8649, 10715, 13504, 13604

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), this sequence (b=15), A029733 (b=16), A118651 (b=17).

p15Q[n_]:=Module[{id15=IntegerDigits[n^2,15]},id15==Reverse[id15]]; Select[ Range[0,14000],p15Q] (* Harvey P. Dale, Jun 03 2020 *)

A263608 Palindromes which are base-3 representations of squares.

Original entry on oeis.org

0, 1, 11, 121, 10201, 11111, 112211, 122221, 1002001, 1120211, 11022011, 100020001, 101212101, 122111221, 1012112101, 1100220011, 10000200001, 10111011101, 110002200011, 111221122111, 1000002000001, 1001221221001, 1012200022101, 1101202021011, 1221221221221, 10101111110101

Offset: 1

N. J. A. Sloane, Oct 22 2015

Robert Israel, Table of n, a(n) for n = 1..143
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A003166, A029984, A262607, A029985.

Intersection of A001738 and A118594.

rev3:= proc(n) local L,i; L:= convert(n,base,3); add(L[-i]*3^(i-1),i=1..nops(L)) end proc:
c3:= proc(n) local L,i; L:= convert(n,base,3); add(L[i]*10^(i-1),i=1..nops(L)) end proc:
R:= 0,1: count:= 2:
for d from 2 while count < 100 do
    if d::odd then
      V:= select(issqr, [seq(seq(a*3^((d+1)/2) + b*3^((d-1)/2)+rev3(a),b=0..2),a=3^((d-3)/2) .. 3^((d-1)/2)-1)])
    else
      V:= select(issqr, [seq(a*3^(d/2) + rev3(a), a=3^(d/2-1) .. 3^(d/2)-1)]);
    fi;
    count:= count+nops(V);
    R:= R, op(map(c3,V));
od:
R; # Robert Israel, May 19 2024

Name edited by Robert Israel, May 19 2024

A263612 Palindromes in base 5 which are also squares.

Original entry on oeis.org

0, 1, 4, 121, 10201, 12321, 114411, 1002001, 1234321, 100020001, 102030201, 121242121, 131141131, 10000200001, 10221412201, 12102420121, 131441144131, 1000002000001, 1002003002001, 1020304030201, 1143442443411, 1210024200121, 4133144413314, 4342230322434, 13431400413431, 100000020000001

Offset: 1

N. J. A. Sloane, Oct 23 2015

Terms displayed in base 5. - Harvey P. Dale, Jan 10 2023

G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A003166, A029988, A262611, A029989.

FromDigits[IntegerDigits[#,5]]&/@Select[Range[0,100000]^2,IntegerDigits[ #,5] == Reverse[ IntegerDigits[ #,5]]&] (* Harvey P. Dale, Jan 10 2023 *)

A229766 Odd numbers whose square's binary reversal is also a square.

Original entry on oeis.org

1, 3, 4523, 11991, 18197, 66075, 72225, 141683, 1092489, 3168099, 6001209, 6226335, 6435309, 12489657, 17906499, 50429883, 51928701, 68301841, 295742437, 390117873, 542959199, 554456167, 566494057

Offset: 1

Alex Ratushnyak, Dec 19 2013

A003166 is a subsequence, except A003166(1)=0.

All odd numbers n such that A068527(A030101(n^2)) = 0. - Antti Karttunen, Dec 20 2013

Cf. A000290, A003166, A030101, A068527, A229687.

a(n) = sqrt(A229687(n)).

cp's OEIS Frontend

A029805 Numbers k such that k^2 is palindromic in base 8.

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A029994 Numbers k such that k^2 is palindromic in base 9.

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A029996 Numbers k such that k^2 is palindromic in base 11.

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A029737 Numbers whose square is palindromic in base 12.

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A029998 Numbers k such that k^2 is palindromic in base 13.

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A030072 Numbers k such that k^2 is palindromic in base 14.

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A030073 Numbers k such that k^2 is palindromic in base 15.

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A263608 Palindromes which are base-3 representations of squares.

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A263612 Palindromes in base 5 which are also squares.

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Mathematica

A229766 Odd numbers whose square's binary reversal is also a square.

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