A002779 -id:A002779 - cp's OEIS frontend

A263616 Number of n-digit numbers whose square is a palindrome.

4, 3, 8, 5, 11, 6, 19, 14, 25, 18, 49, 31, 71, 46, 105, 71, 154, 101, 209, 132, 292, 182, 384, 236, 497, 302, 636, 383, 799, 475, 981, 578, 1201, 701

Offset: 1

N. J. A. Sloane, Oct 23 2015

Number of terms in A002778 with exactly n digits.

			a(2) = 3 because there are three 2-digit numbers with palindromic squares: 11^2 = 121, 22^2 = 484, 26^2 = 676.

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

Cf. A002778, A002779, A263617, A263618, A263619.

Join[{4},Table[Total[Table[If[PalindromeQ[n^2],1,0],{n,10^x,10^(x+1)-1}]],{x,9}]] (* Harvey P. Dale, Apr 09 2019 *)

from itertools import product
def pal(n): s = str(n); return s == s[::-1]
def a(n): return int(n==1) + sum(pal(i**2) for i in range(10**(n-1), 10**n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Apr 03 2021

a(9)-a(10) from Chai Wah Wu, Oct 25 2015
a(11) from Michael S. Branicky, Apr 03 2021
a(12)-a(22) (using A002778) from Chai Wah Wu, Sep 16 2021
a(23)-a(34) from A002778 added by Max Alekseyev, Apr 08 2025

A065379 Palindromic squares with a prime root.

Original entry on oeis.org

4, 9, 121, 10201, 94249, 900075181570009, 12124434743442121, 12323244744232321, 12341234943214321, 1022321210249420121232201, 1210024420147410244200121, 1210222232227222322220121

Offset: 1

Patrick De Geest, Nov 03 2001

Record sporadic palindromic square with a prime root is 186627875420278656872024578726681.

			a(6) = 900075181570009 = p^2 with p = 30001253, a prime.

Patrick De Geest, Palindromic Squares in bases 2 to 17
G. L. Honaker, Jr. and C. Caldwell, Prime Curios!
William Rex Marshall, Palindromic Squares [Internet Archive Wayback Machine]

Cf. A065378, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.

A131760 Numbers n such that n multiplied by its reverse yields a fourth power.

Original entry on oeis.org

1, 4, 9, 121, 484, 676, 2178, 8712, 10000, 10201, 12321, 14641, 40000, 40804, 44944, 69696, 90000, 94249, 698896, 1002001, 1210000, 1234321, 4008004, 4840000, 5221225, 6760000, 6948496, 21780000, 87120000, 100000000, 100020001

Offset: 1

Tanya Khovanova, Sep 17 2007

This sequence contains palindromic squares and palindromic squares with trailing zeros. Are 2178 and 8721 the only non-palindromic reversible pair in this sequence without trailing zeros?

			2178 = 2*9*121 and 8712 = 8*9*121, 2718*8712 = (2*3*11)^4.

Chai Wah Wu, Table of n, a(n) for n = 1..108

Cf. A002779 = Palindromic squares.

Select[Range[1000000], IntegerQ[(#*FromDigits[Reverse[IntegerDigits[ # ]]])^(1/4)] &]

a(20)-a(31) from Donovan Johnson, Oct 27 2008

A176923 Squares of A057148 taken as decimal numbers.

Original entry on oeis.org

0, 1, 121, 10201, 12321, 1002001, 1234321, 100020001, 102030201, 121242121, 123454321, 10000200001, 10221412201, 12102420121, 12345654321, 1000002000001, 1002003002001, 1020304030201, 1022325232201, 1210024200121, 1212225222121, 1232346432321, 1234567654321, 100000020000001, 100220141022001

Offset: 1

Jeremy Gardiner, Apr 29 2010

See comment in A057148.

Cf. A002779, A057148, A063009.

def A176923(n):
    if n == 1: return 0
    a = 1<Chai Wah Wu, Jun 10 2024

A225739 Palindromic squares whose sum of digits is also a palindromic square.

Original entry on oeis.org

1, 4, 9, 121, 10201, 12321, 1002001, 100020001, 102030201, 10000200001, 1000002000001, 1002003002001, 100000020000001, 10000000200000001, 10002000300020001, 1000000002000000001, 100000000020000000001, 100002000030000200001

Offset: 1

Jayanta Basu, May 14 2013

Are there finitely many terms not of the form (10^n+1)^2 or (100^n+10^n+1)^2? I haven't found any. - Charles R Greathouse IV, May 14 2013

			12321 is included because it is a palindromic square and 1+2+3+2+1=9 is also a palindromic square.
5265533355625 is not included because although it is a palindromic square its sum of digits, 55, is not.

Subsequence of A002779.

id[n_]:=IntegerDigits[n]; palQ[n_]:=Reverse[id[n]]==id[n]; t={}; Do[If[palQ[x=n^2] && palQ[y=Total[id[x]]] && IntegerQ[Sqrt[y]], AppendTo[t,x]],{n,1.2*10^6}]; t

ispal(n)=my(v=digits(n));for(i=1,#v\2,if(v[i]!=v[#v+1-i],return(0)));1
for(n=1,1e6,s=sumdigits(n^2); issquare(s) && ispal(s) && ispal(n^2) && print1(n^2", ")) \\ Charles R Greathouse IV, May 14 2013

a(n) < 32^n. - Charles R Greathouse IV, May 14 2013

a(13)-a(18) from Charles R Greathouse IV, May 14 2013

A343098 Number of palindromes < 10^n whose squares are also palindromes.

Original entry on oeis.org

1, 4, 6, 11, 14, 22, 27, 40, 49, 71, 87, 124, 151, 211, 254, 347, 412, 550, 644, 841, 972, 1244, 1421, 1786, 2019, 2497, 2797, 3410, 3789, 4561, 5032, 5989, 6566, 7736, 8434, 9847, 10682, 12370, 13359, 15356, 16517, 18859, 20211, 22936, 24499, 27647, 29442, 33055

Offset: 0

Chai Wah Wu, Apr 04 2021

Partial sum of A218035. Number of terms in A057135 < 10^n.

			a(2) = 6 since the only palindromes < 100 whose square are palindromes are 0,1,2,3,11,22.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A000290, A002113, A002779, A057136, A057135, A218035.

A343098_list = [1, 4, 6, 11, 14, 22, 27, 40, 49, 71]
for i in range(10**3): A343098_list.append(A343098_list[-1] + 4*A343098_list[-2] - 4*A343098_list[-3] - 6*A343098_list[-4] + 6*A343098_list[-5] + 4*A343098_list[-6] - 4*A343098_list[-7] - A343098_list[-8] + A343098_list[-9])

a(n) = #{i:A057135(i)<10^n}.
For n > 0, a(n) = Sum_{i=1..n} A218035(i).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: (-x^9 + x^7 - x^6 - 6*x^5 - x^4 + 7*x^3 + 2*x^2 - 3*x - 1)/((x - 1)^5*(x + 1)^4).
a(n) = 1491 + 904*n + 510*n^2 - 52*n^3 + 6*n^4 + (-1)^n * (45 - 296*n + 42*n^2 - 4*n^3) for n>0. - Greg Dresden, Jun 20 2021

A358237 Palindromes of the form k^2 + 2.

Original entry on oeis.org

2, 3, 6, 11, 66, 171, 363, 7227, 66566, 154451, 685586, 3279723, 6441446, 8503058, 8916198, 13155131, 20611602, 110313011, 19833933891, 72288388227, 89064046098, 179298892971, 814860068418, 1103556553011, 3406132316043, 6205240425026, 30403655630403, 206380232083602, 666150525051666

Offset: 1

Robert Xiao, Nov 04 2022

Cf. A059100, A358207, A070254, A002779.

Select[Range[0,26*10^6]^2+2,PalindromeQ] (* Harvey P. Dale, Sep 01 2024 *)

A370512 Largest palindromic square which is a concatenation of partitions of n; or 0 if no such number exists.

Original entry on oeis.org

1, 0, 0, 121, 0, 0, 0, 0, 12321, 0, 0, 0, 121, 0, 0, 121242121, 0, 12321, 5221225, 0, 0, 121, 0, 0, 1212225222121, 0, 12321, 5221225, 0, 0, 10201, 0, 0, 1212225222121, 0, 12122232623222121

Offset: 1

Chai Wah Wu, Feb 20 2024

			Note that a(4) = a(13) = a(22) = 121 as the digits of 121 can be partitioned as 1+2+1 or 12+1 or 1+21.

Cf. A002779, A079842.

from collections import Counter
from operator import itemgetter
from sympy.ntheory.primetest import is_square
from sympy.utilities.iterables import partitions, multiset_permutations
def A370512(n):
    smax, m = 0, 0
    for s, p in sorted(partitions(n,size=True),key=itemgetter(0),reverse=True):
        if s0:
                smax=s
    return m

a(n) <= A079842(n).

If n is a palindromic square, then a(n) >= n.

A087369 Smallest palindromic square multiple of the n-th palindrome.

Original entry on oeis.org

1, 4, 9, 4, 5221225, 69696, 1002001, 44944, 9, 121, 484, 69696, 484, 522666037791100950480675576084059001197730666225, 69696, 1002001, 69696, 69696, 10201, 12321, 121, 12100022220012621002222000121, 12122010222622201022121, 123230205292502032321, 102012022050220210201, 94206450305460249, 1000220232126212320220001, 12104402820440121, 40804, 44944

Offset: 1

Amarnath Murthy, Sep 08 2003

			a(5) = 5221225 = 2285^2.

Subset of A002779.

Cf. A002778.

a(7)-a(13) from Sean A. Irvine, Mar 01 2010

Terms a(14) onward from Max Alekseyev, Apr 02 2025

A087988 Palindromic numbers whose squares and cubes are equally palindromic.

Original entry on oeis.org

0, 1, 2, 11, 101, 111, 1001, 10001, 10101, 11011, 100001, 101101, 110011, 1000001, 1001001, 1100011, 10000001, 10011001, 10100101, 11000011, 100000001, 100010001, 100101001, 101000101, 110000011, 1000000001, 1000110001

Offset: 1

Labos Elemer, Oct 01 2003

Numbers n such that n, n^2 and n^3 are all palindromes.

Essentially A002780 with two terms removed, 7 and 2201.

			11^2=121, 11^3=1331.

Cf. A002779, A002781.

Intersection of A002113, A002778 and A002780.

rev:=proc(a) local aa,ct: aa:=convert(a,base,10): ct:=nops(aa): add(10^(ct-j)*aa[j],j=1..ct) end: p:=proc(n) if rev(n)=n and rev(n^2)=n^2 and rev(n^3)=n^3 then n else fi end: seq(p(n),n=0..12*10^5); # Emeric Deutsch, May 01 2005

ispal(n) = my(d = digits(n)); Vecrev(d) == d;
isok(n) = ispal(n) && ispal(n^2) && ispal(n^3); \\ Michel Marcus, Oct 25 2015

More terms from Ray Chandler, Oct 05 2003

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A263616 Number of n-digit numbers whose square is a palindrome.

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A065379 Palindromic squares with a prime root.

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A131760 Numbers n such that n multiplied by its reverse yields a fourth power.

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A176923 Squares of A057148 taken as decimal numbers.

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A225739 Palindromic squares whose sum of digits is also a palindromic square.

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A343098 Number of palindromes < 10^n whose squares are also palindromes.

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A358237 Palindromes of the form k^2 + 2.

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A370512 Largest palindromic square which is a concatenation of partitions of n; or 0 if no such number exists.

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Python

Formula

A087369 Smallest palindromic square multiple of the n-th palindrome.