A002778 -id:A002778 - cp's OEIS frontend

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

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 *)

A263618 Number of palindromic squares with exactly n digits.

Original entry on oeis.org

4, 0, 3, 0, 7, 1, 5, 0, 11, 0, 5, 1, 19, 0, 13, 1, 25, 0, 18, 0, 48, 1, 31, 0, 70, 1, 44, 2, 105, 0, 70, 1, 153, 1, 98, 3, 209, 0, 132, 0, 291, 1, 181, 1, 384, 0, 234, 2, 496, 1, 301, 1, 636, 0, 383, 0, 798, 1, 474, 1, 981, 0, 578, 0, 1199, 2, 701

Offset: 1

N. J. A. Sloane, Oct 23 2015

Number of terms in A002779 with exactly n digits.
a(24) = a(30) = a(38) = a(40) = 0. - Robert Price, Apr 26 2019
a(2*k+1) > 0 since (10^k+1)^2 is a palindrome of 2*k+1 digits. - Chai Wah Wu, Jun 14 2024

G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy] See page 95.

Cf. A002778, A002779, A263617, A263617, A263619, A263620.

Cf. A034822 (positions of zeros).

Table[Length[Select[Range[If[n == 1, 0, Ceiling[Sqrt[10^(n - 1)]]],Floor[Sqrt[10^n]]], #^2 == IntegerReverse[#^2] &]], {n, 15}] (* Robert Price, Apr 26 2019 *)

a(13)-a(19) from Chai Wah Wu, Oct 24 2015
a(20) from Robert Price, Apr 26 2019
a(21)-a(44) (using A002778) from Chai Wah Wu, Sep 16 2021
a(45)-a(67) from A002778 added by Max Alekseyev, Apr 08 2025

A059744 Numbers k such that k^2 is a palindromic square of sporadic type.

Original entry on oeis.org

26, 264, 307, 836, 2285, 2636, 22865, 24846, 30693, 798644, 1042151, 1270869, 2012748, 2294675, 3069307, 11129361, 12028229, 12866669, 30001253, 64030648, 306930693, 2062386218, 2481623254, 10106064399, 10207355549, 13579355059, 22865150135, 30101273647, 30693069307

Offset: 1

N. J. A. Sloane, Feb 21 2001

C. Ashbacher, More on palindromic squares, J. Rec. Math. 22, no. 2 (1990), 133-135. [A scan of the first page of this article is included with the last page of the Keith (1990) scan]
J. K. R. Barnett, "Tables of Square Palindromes in Bases 2 and 10," Journal of Recreational Mathematics, 23:1, pp. 13-18, 1991.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 26, pp 10, Ellipses, Paris 2008.
M. Keith, "Classification and Enumeration of Palindromic Squares," Journal of Recreational Mathematics, 22:2, pp. 124-132, 1990.
R. Ondrejka, "A Palindrome (151) of Palindromic Squares," Journal of Recreational Mathematics, 20:1, pp. 68-71, 1988.

Patrick De Geest, Full List of the Sporadic Square Palindromes (SSP), part of Palindromic Squares website on WorldOfNumbers.
M. Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy] (Has ten more terms)
W. R. Marshall, Table up to 10^20

Cf. A059745, A016106, A028818, A060087.

Select[Range[1042151], ! PalindromeQ[#] && PalindromeQ[#^2] &] (* Michael De Vlieger, Oct 03 2023, not suitable for terms > 1042151, needs amendment for larger terms *)

More terms from WorldOfNumbers website, communicated by Hugo Pfoertner, Oct 03 2023

A102859 Numbers that when squared and written backwards give a square again.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 26, 30, 31, 33, 99, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 130, 200, 201, 202, 210, 211, 212, 220, 221, 260, 264, 300, 301, 307, 310, 311, 330, 836, 990, 1000, 1001, 1002, 1003, 1010, 1011, 1012, 1013, 1020

Offset: 1

Sanita Kashcheyeva (sanits(AT)gmail.com), Mar 01 2005

Contains A002778. - Robert Israel, Sep 20 2015

Squares of these terms are in A061457. - Jon E. Schoenfield, May 17 2022

			a(7)=12 belongs to the sequence since writing 12^2 = 144 backwards gives 441 = 21^2.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
Index entry for sequences related to reversing digits of squares

Cf. A002778, A002942.

Cf. A061457 (squares).

[n: n in [0..1100] | IsSquare(Seqint(Reverse(Intseq(n^2))))]; // Vincenzo Librandi, Sep 21 2015

rev:= proc(n)
  local L, Ln, i;
  L:= convert(n, base, 10);
  Ln:= nops(L);
  add(L[i]*10^(Ln-i), i=1..Ln);
end proc:
select(t -> issqr(rev(t^2)),[$0..10^5]); # Robert Israel, Sep 20 2015

Select[Range[1000], IntegerQ[Sqrt[FromDigits[Reverse[IntegerDigits[ #^2]]]]] &]

from itertools import count, islice
from sympy import integer_nthroot
def A102859_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:integer_nthroot(int(str(n**2)[::-1]),2)[1], count(max(startvalue,0)))
A102859_list = list(islice(A102859_gen(),30)) # Chai Wah Wu, Nov 18 2022

a(n) = sqrt(A061457(n)). - Jon E. Schoenfield, May 17 2022

0 inserted by Jon E. Schoenfield, Sep 20 2015

A002781 Palindromic cubes.

Original entry on oeis.org

0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, 10662526601, 1000300030001, 1030607060301, 1334996994331, 1000030000300001, 1033394994933301, 1331399339931331, 1000003000003000001, 1003006007006003001, 1331039930399301331

Offset: 1

N. J. A. Sloane

a(9) = 1066252601 = 2201^3 is the unique known palindromic cube that has a non-palindromic rootnumber (see comments in A002780 and Penguin reference). - Bernard Schott, Oct 21 2021

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 10662526601, page 188.

T. D. Noe, Table of n, a(n) for n = 1..89 (from De Geest)
Patrick De Geest, Palindromic Cubes
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
G. J. Simmons, Palindrome cubes: Problem B-183, Fibonacci Quart. 8 (1970), no. 5, p. 551.

Cf. A002780.

Intersection of A000578 and A002113.

Select[Range[0,12*10^5]^3,PalindromeQ[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 02 2017 *)

ispal(x) = my(d=digits(x)); d == Vecrev(d); \\ A002113
lista(nn) = my(list = List(), c); for (n=0, sqrtnint(nn, 3), if (ispal(c=n^3), listput(list, c));); Vec(list); \\ Michel Marcus, Oct 21 2021

a(n) = A002780(n)^3.

Thanks to Pierre Genix (Pierre.Genix(AT)wanadoo.fr) and Harvey P. Dale who pointed out that there were errors in earlier versions of this sequence.

A007573 a(n) is the number of base numbers with 2n+1 digits in the asymmetric families of palindromic squares.

Original entry on oeis.org

1, 2, 5, 6, 9, 10, 10, 15, 15, 16, 18, 24, 18, 26, 24, 30, 27, 33, 28, 40, 33, 40, 35, 48, 37, 50, 42, 53, 45, 58, 46, 64, 50, 64, 54, 72, 55, 73, 60, 78, 63, 82, 63, 88, 69, 88, 72, 95, 73, 98, 78, 102, 80, 106, 82, 112, 87, 111, 90, 120, 91, 122, 95, 126, 99, 130, 100, 135

Offset: 3

Simon Plouffe, N. J. A. Sloane, Robert G. Wilson v

			a(3) = 1: The only base number of length 2*3 + 1 = 7 is 1109111 = A060087(1);
a(4) = 2 indicates the existence of two length 2*4 + 1 = 9 base numbers, 110091011 = A060087(2) and 111091111 = A060087(3).

M. Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 3..115
Patrick De Geest, Subsets of Palindromic Squares
IBM Research Ponder This, Non-palindromic numbers with palindromic squares, October 2023 - Challenge.
M. Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]. See foot of page 130.
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A002779, A060087, A060088.

PARI
```
\\ see P. De Geest link.
```

a(17)-a(31) from Sean A. Irvine, Jan 10 2018
Name and offset corrected by Hugo Pfoertner, Oct 04 2023
a(32)-a(70) from Hugo Pfoertner, Oct 07 2023

A016113 Numbers whose square is a palindrome with an even number of digits.

Original entry on oeis.org

836, 798644, 64030648, 83163115486, 6360832925898, 69800670077028, 98275825201587, 6819209882215742, 40447213778058769, 404099764753665981, 633856150760638652, 795559265009384106, 637323988797048057098, 3823177109095314778621

Offset: 1

Robert G. Wilson v

For the squares, see A027829(n) = a(n)^2. - M. F. Hasler, Oct 11 2019

C. Ashbacher, More on palindromic squares, J. Rec. Math. 22, no. 2 (1990), 133-135. [A scan of the first page of this article is included with the last page of the Keith (1990) scan]

Max Alekseyev, Table of n, a(n) for n = 1..22 (from Patrick De Geest's website)
K. S. Brown, On General Palindromic Numbers
Patrick De Geest, Palindromic Squares in bases 2 to 17
P. De Geest, Subsets of Palindromic Squares
M. Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]
F. Yuan, Palindromic Square Numbers, as of July 2002.

A proper subset of A002778.

Cf. A027829.

is_A016113(n)={Vecrev(n=digits(n^2))==n&&!bittest(#n,0)} \\ This is faster than first checking for even length, if applied to numbers in a range where the squares are known to have an even number of digits, as should be the case for a systematic search. - M. F. Hasler, Jun 08 2014

Two terms were found by Bennett from UK (communication from Patrick De Geest)
Edited by M. F. Hasler, Jun 08 2014
Missing a(10) inserted by M. F. Hasler, Oct 11 2019

A027719 Numbers k such that k^2 + 1 is a palindrome.

Original entry on oeis.org

0, 1, 2, 10, 25, 100, 1000, 1020, 1489, 2248, 10000, 10090, 100000, 100910, 102020, 167491, 1000000, 1000200, 1009090, 2744934, 10000000, 10000900, 10090910, 24917195, 100000000, 100909090, 103226660, 271867456, 1000000000, 1000002000, 1009090910, 1577033471

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..48
P. De Geest, Palindromic incremented squares of the form n^2+1

Cf. A002522, A027720, A002778, A070253.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; Select[Range[0, 10^5], palQ[#^2 + 1] &] (* Giovanni Resta, Aug 29 2018 *)

More terms from Giovanni Resta, Aug 28 2018

cp's OEIS Frontend

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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A263618 Number of palindromic squares with exactly n digits.

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A059744 Numbers k such that k^2 is a palindromic square of sporadic type.

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A102859 Numbers that when squared and written backwards give a square again.

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A002781 Palindromic cubes.

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A007573 a(n) is the number of base numbers with 2n+1 digits in the asymmetric families of palindromic squares.

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