A002779 -id:A002779 - cp's OEIS frontend

A028504 Palindromes of form k*(k+2); or palindromes 1 less than a square.

0, 3, 8, 99, 323, 575, 4224, 5775, 9999, 36863, 42024, 999999, 3055503, 3640463, 5597955, 8803088, 32855823, 99999999, 360696063, 422919224, 9999999999, 30485858403, 30536863503, 32154945123, 59080108095, 86310801368, 304816618403, 999999999999, 3490500050943

Offset: 1

Patrick De Geest

10^(2*m) - 1 for m > 0 are terms. - Chai Wah Wu, May 25 2017

			4224 belongs to this sequence as 4225 = 65^2.

Giovanni Resta, Table of n, a(n) for n = 1..53
P. De Geest, Palindromic quasipronic numbers of the form n(n+2)

Cf. A005563, A028503, A002779, A070253, A070254.

stop := 400000; m := 1; while m < stop do s := m*m - 1; if s = int_reverse(s) then write(s," "); end; inc(m); end;

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; Select[Range[10000]^2 - 1, palQ] (* Giovanni Resta, Aug 29 2018 *)
Select[Table[n(n+2),{n,0,19*10^5}],PalindromeQ] (* Harvey P. Dale, Oct 13 2024 *)

a(n) = A028503(n) * (A028503(n) + 2) = A070253(n)^2 - 1 = A070254(n) - 1. - Giovanni Resta, Aug 29 2018

More terms from Giovanni Resta, Aug 28 2018

A027829 Palindromic squares with an even number of digits.

Original entry on oeis.org

698896, 637832238736, 4099923883299904, 6916103777337773016196, 40460195511188111559106404, 4872133543202112023453312784, 9658137819052882509187318569, 46501623417708833880771432610564, 1635977102407987117897042017795361, 163296619873968186681869378916692361

Offset: 1

Keith Devlin, via Boon Leong (boon_leong(AT)hotmail.com)

			836^2 = 698896, which is palindromic, so 698896 is in the sequence.
1001^2 = 1002001, which is palindromic, but it has an odd number of digits, so it's not in the sequence.

Charles 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]

M. F. Hasler, Table of n, a(n) for n = 1..14
K. S. Brown, On General Palindromic Numbers
Patrick De Geest, Sporadic palindromic squares of even length
Michael Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]

Cf. A002113, A002778, A002779, A016113.

Select[Range[1000000]^2, PalindromeQ[#] && OddQ[Floor[Log[10, #]]] &] (* Alonso del Arte, Oct 11 2019 *)

is_A027829(n)={issquare(n)&&Vecrev(n=digits(n))==n&&!bittest(#n, 0)} \\ This is faster than first checking for even length if applied to numbers known to have an even number of digits, as should be the case for a systematic search. For this, one should only consider squares, i.e., rather use is_A016113.  - M. F. Hasler, Jun 08 2014

from math import isqrt
from itertools import count, islice
def A027829_gen(): # generator of terms
    return filter(lambda n: (s:=str(n))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1], map(lambda n: n**2, (d for l in count(2,2) for d in range(isqrt(10**(l-1))+1,isqrt(10**l)+1))))
A027829_list = list(islice(A027829_gen(),3)) # Chai Wah Wu, Jun 23 2022

def isPalindromic(n: BigInt): Boolean = n.toString == n.toString.reverse
val squares = ((1: BigInt) to (1000000: BigInt)).map(n => n * n)
squares.filter(n => isPalindromic(n) && n.toString.length % 2 == 0) // Alonso del Arte, Oct 07 2019

a(n) = A016113(n)^2. - M. F. Hasler, Jun 08 2014

Two new terms were recently found by Bennett from UK (communication from Patrick De Geest, Dec 1999 or before)

Edited by M. F. Hasler, Jun 08 2014

A180436 Palindromic numbers which are sum of consecutive squares.

Original entry on oeis.org

1, 4, 5, 9, 55, 77, 121, 181, 313, 434, 484, 505, 545, 595, 636, 676, 818, 1001, 1111, 1441, 1771, 4334, 6446, 10201, 12321, 14641, 17371, 17871, 19691, 21712, 40804, 41214, 42924, 44444, 44944, 46564, 51015, 65756, 69696, 81818, 94249, 97679, 99199

Offset: 1

Zhining Yang, Jan 19 2011

In more than one way: 554455, 9343439, ... (A267600) - Robert G. Wilson v, May 28 2012

			1001 is in the sequence because 1001 is palindromic and it can be written as sum of consecutive squares (1001 = 4^2 + 5^2 + 6^2 + ... + 13^2 + 14^2).

Giovanni Resta, Table of n, a(n) for n = 1..10400 (terms < 10^18, first 228 terms from Robert G. Wilson v)

Cf. A002113, A002779, A034705, A216446, A267600.

palQ[n_Integer] := Block[{idn = IntegerDigits[n]}, idn == Reverse[idn]]; lst = {}; k = 1; While[k < 1000, AppendTo[lst, Select[ Accumulate[ Range[k, 1000]^2], palQ]]; lst = Union@ Flatten@ lst; k++]; Select[lst, # < 10^6 &] (* Robert G. Wilson v, May 28 2012 *)

A186080 Fourth powers that are palindromic in base 10.

Original entry on oeis.org

0, 1, 14641, 104060401, 1004006004001, 10004000600040001, 100004000060000400001, 1000004000006000004000001, 10000004000000600000040000001, 100000004000000060000000400000001, 1000000004000000006000000004000000001, 10000000004000000000600000000040000000001, 100000000004000000000060000000000400000000001

Offset: 1

Matevz Markovic, Feb 11 2011

See A056810 (the main entry for this problem) for further information, including the search limit. - N. J. A. Sloane, Mar 07 2011
Conjecture: If k^4 is a palindrome > 0, then k begins and ends with digit 1, all other digits of k being 0.
The number of zeros in 1x1, where the x are zeros, is the same as (the number of zeros)/4 in (1x1)^4 = 1x4x6x4x1.

P. De Geest, Palindromic cubes (The Simmons test is mentioned here) [broken link]
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]

Cf. A002113, A168576, A056810, A002778, A002779.

[ p: n in [0..10000000] | s eq Reverse(s) where s is Intseq(p) where p is n^4 ];

Do[If[Module[{idn = IntegerDigits[n^4, 10]}, idn == Reverse[idn]], Print[n^4]], {n, 100000001}]

a(n) = A056810(n)^4.

a(11)-a(13) using extensions of A056810 from Hugo Pfoertner, Oct 22 2021

A263617 Number of numbers with at most n digits whose square is a palindrome.

Original entry on oeis.org

4, 7, 15, 20, 31, 37, 56, 70, 95, 113, 162, 193, 264, 310, 415, 486, 640, 741, 950, 1082, 1374, 1556, 1940, 2176, 2673, 2975, 3611, 3994, 4793, 5268, 6249, 6827, 8028, 8729

Offset: 1

N. J. A. Sloane, Oct 23 2015

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

Partial sums of A263616.

Cf. A002778, A002779, A263618, A263619.

Table[Length[Select[Range[ 0, 10^n - 1], PalindromeQ[#^2] &]], {n, 6}] (* Robert Price, Apr 26 2019 *)

a(9)-a(10) from Chai Wah Wu, Oct 25 2015
a(11)-a(12) from Michael S. Branicky, May 23 2021
a(13)-a(22) (using A002778) from Chai Wah Wu, Sep 16 2021
a(23)-a(34) from Max Alekseyev, Apr 08 2025

A263619 Number of palindromic squares with at most n digits.

Original entry on oeis.org

4, 4, 7, 7, 14, 15, 20, 20, 31, 31, 36, 37, 56, 56, 69, 70, 95, 95, 113, 113, 161, 162, 193, 193, 263, 264, 308, 310, 415, 415, 485, 486, 639, 640, 738, 741, 950, 950, 1082, 1082, 1373, 1374, 1555, 1556, 1940, 1940, 2174, 2176, 2672, 2673, 2974, 2975, 3611, 3611, 3994, 3994, 4792, 4793, 5267, 5268, 6249, 6249, 6827, 6827, 8026, 8028, 8729

Offset: 1

N. J. A. Sloane, Oct 23 2015

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

Partial sums of A263618.

Cf. A002778, A002779, A263616, A263617, A263620.

Table[Length[Select[Range[0, Floor[Sqrt[10^n]]], PalindromeQ[#^2] &]], {n, 10}] (* 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 A263618 from Chai Wah Wu, Jun 14 2024
a(45)-a(67) added by Max Alekseyev, Apr 08 2025

A307717 Number of palindromic squares, k^2, of length n such that k is also palindromic.

Original entry on oeis.org

4, 0, 2, 0, 5, 0, 3, 0, 8, 0, 5, 0, 13, 0, 9, 0, 22, 0, 16, 0, 37, 0, 27, 0, 60, 0, 43, 0, 93, 0, 65, 0, 138, 0, 94, 0, 197, 0, 131, 0, 272, 0, 177, 0, 365, 0, 233, 0, 478, 0, 300, 0, 613, 0, 379, 0, 772, 0, 471, 0, 957, 0, 577, 0, 1170, 0, 698, 0, 1413, 0

Offset: 1

Robert Price, Apr 23 2019

			There are only two palindromic squares of length 3 whose root is also palindromic. 11^2=121 and 22^2=484. Thus, a(3)=2.

Patrick De Geest, Palindromic Squares in bases 2 to 17
M. Kauers and C. Koutschan, Guessing with Little Data, arXiv:2202.07966 [cs.SC], 2022.
Bertrand Teguia Tabuguia and Wolfram Koepf, FPS In Action: An Easy Way To Find Explicit Formulas For Interlaced Hypergeometric Sequences, arXiv:2207.01031 [cs.SC], 2022.
Bertrand Teguia Tabuguia, Hypergeometric-Type Sequences, arXiv:2401.00256 [cs.SC], 2023.
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Cf. A002778, A002779, A034307, A034822, A218035.

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

From Christoph Koutschan, Feb 19 2022: (Start)
a(2n-1) = A218035(n).
a(n) is given by a quasi-polynomial (for a proof, see A218035):
a(1) = 4;
a(2n) = 0;
a(4n+1) = (n^3-3*n^2+11*n+6)/3 (n > 0);
a(4n+3) = (n^3+5*n+12)/6 (n >= 0). (End)

a(16)-a(20) from Robert Price, Apr 25 2019

a(21)-a(70) from Giovanni Resta, Apr 28 2019

A065378 Primes p such that p^2 is a palindromic square.

Original entry on oeis.org

2, 3, 11, 101, 307, 30001253, 100111001, 110111011, 111010111, 111091111, 1011099011101, 1100011100011, 1100101010011, 1101010101011, 100110101011001, 100110990111001, 101000010000101, 101011000110101, 101110000011101

Offset: 1

Patrick De Geest, Nov 03 2001

Record prime base number of a sporadic palindromic square is 13661181333262459.

			E.g. a(6) = 900075181570009 = p^2 with p = 30001253 and prime.

Hans Havermann (via Feng Yuan), Table of n, a(n) for n = 1..61
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. L. Honaker, Jr. and C. Caldwell, Prime Curios!
W. R. Marshall, Palindromic Squares
F. Yuan, Palindromic Square Numbers

Cf. A065379, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.

t={}; Do[p=Prime[n]; If[FromDigits[Reverse[IntegerDigits[p^2]]]==p^2,AppendTo[t,p]],{n,10^7}]; t (* Jayanta Basu, May 11 2013 *)

Corrected by Jayanta Basu, May 11 2013

A263620 Number of nonzero palindromic squares with at most 2n digits.

Original entry on oeis.org

3, 6, 14, 19, 30, 36, 55, 69, 94, 112, 161, 192, 263, 309, 414, 485, 639, 740, 949, 1081, 1373, 1555

Offset: 1

N. J. A. Sloane, Oct 23 2015

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, A263618, A263619.

Table[Length[Select[Range[1, Floor[Sqrt[10^(2 n)]]], PalindromeQ[#^2] &]], {n, 6}] (* Robert Price, Apr 26 2019 *)

def ispal(n): s = str(n); return s == s[::-1]
def a(n):
  c, k, kk = 0, 1, 1
  while kk < 10**(2*n): c, k, kk = c + (ispal(kk)), k+1, kk + 2*k + 1
  return c
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Mar 06 2021

Equals A263619(2n)-1.

a(7)-a(10) from Chai Wah Wu, Oct 25 2015

More terms using A263618 from Chai Wah Wu, Jun 14 2024

A028817 Palindromic squares with an odd number of digits.

Original entry on oeis.org

1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, 69696, 94249, 1002001, 1234321, 4008004, 5221225, 6948496, 100020001, 102030201, 104060401, 121242121, 123454321, 125686521, 400080004, 404090404, 522808225, 617323716, 942060249, 10000200001

Offset: 1

Patrick De Geest

Patrick De Geest, Subsets of Palindromic Squares

Cf. A002779, A028816.

id[n_] := IntegerDigits[n]; palQ[n_] := FromDigits[Reverse[id[n]]] == n; t = {}; Do[If[palQ[x = n^2] && OddQ[Length[id[x]]], AppendTo[t, x]],{n, 101000}]; t (* Jayanta Basu, May 13 2013 *)
Select[Range[100000]^2, PalindromeQ[#] && EvenQ[Floor[Log[10, #]]] &] (* Alonso del Arte, Oct 11 2019 *)

def isPalindromic(n: BigInt): Boolean = n.toString == n.toString.reverse
val squares = ((1: BigInt) to (1000000: BigInt)).map(n => n * n)
squares.filter(n => isPalindromic(n) && n.toString.length % 2 == 1) // Alonso del Arte, Oct 07 2019

cp's OEIS Frontend

A028504 Palindromes of form k*(k+2); or palindromes 1 less than a square.

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A027829 Palindromic squares with an even number of digits.

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A180436 Palindromic numbers which are sum of consecutive squares.

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A186080 Fourth powers that are palindromic in base 10.

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A263617 Number of numbers with at most n digits whose square is a palindrome.

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A263619 Number of palindromic squares with at most n digits.

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A307717 Number of palindromic squares, k^2, of length n such that k is also palindromic.

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