A002778 -id:A002778 - cp's OEIS frontend

A175440 Terms in A177950 that are not in A002778.

9, 33, 48, 66, 87, 99, 117, 216, 273, 288, 297, 333, 484, 513, 528, 666, 783, 819, 999, 1323, 1331, 1452, 1474, 1602, 2178, 2622, 3333, 4884, 4961, 6666, 7161, 7575, 9999, 10989, 11979, 12969, 14652, 14733, 15972, 20402, 21021, 21534, 21648, 23331

Offset: 1

Zak Seidov, May 16 2010

Terms n in A177950 such that n^2 is palindrome are also in A002778.
This sequence consists of numbers (more interesting?) that are in A177950 but not in A002778.
Notice that most terms are multiples of 3, while some are not: 484, 1331, 1474, 4961, 20402, 48884, 122221, 188887, 217822, 467126, 477773, 484484, 506506, 525503, 718189, 808808, 1461262, 1729408, 2004002, 2317315, 2920819, 4840484.

Zak Seidov, A175440

Cf. A002778 Numbers whose square is a palindrome, A177950 Numbers n dividing n^2 with digits reversed.

A002779 Palindromic squares.

Original entry on oeis.org

0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, 69696, 94249, 698896, 1002001, 1234321, 4008004, 5221225, 6948496, 100020001, 102030201, 104060401, 121242121, 123454321, 125686521, 400080004, 404090404, 522808225

Offset: 1

N. J. A. Sloane

These are numbers that are both squares (see A000290) and palindromes (see A002113).

			676 is included because it is both a perfect square and a palindrome.

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).

Hans Havermann (via Feng Yuan), T. D. Noe (from P. De Geest) [to 485], Table of n, a(n) for n = 1..1940
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012-2014. - From _N. J. A. Sloane_, Nov 08 2012
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
Patrick De Geest, Palindromic Squares in bases 2 to 17
W. R. Marshall, Palindromic Squares
Phakhinkon Phunphayap, Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018.
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]
Eric Weisstein's World of Mathematics, Palindromic Number.
Feng Yuan, Palindromic Square Numbers

Cf. A000290, A002113, A002778, A010052, A027829 (subsets), A028817, A029734.
Cf. A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993.
Cf. A029995, A029997, A029999, A030074, A030075, A057136, A136522.

a002779 n = a002778_list !! (n-1)
a002779_list = filter ((== 1) . a136522) a000290_list
-- Reinhard Zumkeller, Oct 11 2011

[k^2:k in [0..100000]| Intseq(k^2) eq Reverse(Intseq(k^2)) ]; // Marius A. Burtea, Oct 15 2019

palindromicNumberQ = ((# // IntegerDigits // Reverse // FromDigits) == #) &; Select[Table[n^2, {n, 0, 9999}],  palindromicNumberQ] (* Herman Beeksma, Jul 14 2005 *)
pb10Q[n_] := Module[{idn10 = IntegerDigits[n, 10]}, idn10 == Reverse[idn10]]; Select[Range[0, 19999]^2, pb10Q] (* Vincenzo Librandi, Jul 24 2014 *)
Select[Range[0, 22999]^2, PalindromeQ] (* Requires Mathematica version 10 or later. - Harvey P. Dale, May 01 2017 *)

is(n)=my(d=digits(n)); d==Vecrev(d) && issquare(n) \\ Charles R Greathouse IV, Feb 06 2017

A002779_list = [int(s) for s in (str(m**2) for m in range(10**5)) if s == s[::-1]] # Chai Wah Wu, Aug 26 2021

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

From Reinhard Zumkeller, Oct 11 2011: (Start)
a(n) = A002778(n)^2.
A136522(A000290(a(n))) = 1.
A010052(a(n)) * A136522(a(n)) = 1. (End)

A003166 Numbers whose square in base 2 is a palindrome.

Original entry on oeis.org

0, 1, 3, 4523, 11991, 18197, 141683, 1092489, 3168099, 6435309, 12489657, 17906499, 68301841, 295742437, 390117873, 542959199, 4770504939, 17360493407, 73798050723, 101657343993, 107137400475, 202491428745, 1615452642807, 4902182461643, 9274278357017, 12863364360297

Offset: 1

N. J. A. Sloane, R. H. Hardin

Numbers k such that k^2 is in A006995.

The only palindromes in this sequence are 0, 1, and 3. See AMM problem 11922. - Max Alekseyev, Oct 22 2022

			3^2 = 9 = 1001_2, a palindrome.
4523^2 = 20457529 = 1001110000010100000111001_2.

G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Don Knuth, Table of n, a(n) for n = 1..50 [This table extends earlier work of Gus Simmons, Jon Schoenfield, Don Knuth, and Michael Coriand]
M. A. Alekseyev, Problem 11922. American Mathematical Monthly 123:7 (2016), 722.
Patrick De Geest, Palindromic Squares in bases 2 to 17
Carlos Rivera, Problem 89. Palindromic binary expression of primes squared, The Prime Puzzles & Problems Connection.
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778 (base 10 analog), A029983 (the actual squares). In binary: A262595, A262596.

Cf. A006995.

Do[c = RealDigits[n^2, 2][[1]]; If[c == Reverse[c], Print[n]], {n, 0, 10^9}]

is(n)=my(b=binary(n^2)); b==Vecrev(b) \\ Charles R Greathouse IV, Feb 07 2017

from itertools import count, islice
def A003166_gen(): # generator of terms
    return filter(lambda k: (s:=bin(k**2)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A003166_list = list(islice(A003166_gen(),10)) # Chai Wah Wu, Jun 23 2022

a(16) = 4770504939 found by Patrick De Geest, May 15 1999
a(17)-a(31) from Jon E. Schoenfield, May 08 2009
a(32) = 285000288617375,
a(33) = 301429589329949,
a(34) = 1178448744881657 from Don Knuth, Jan 28 2013 [who doublechecked the previous results and searched up to 2^104]

A029984 Numbers k such that k^2 is palindromic in base 3.

Original entry on oeis.org

0, 1, 2, 4, 10, 11, 20, 22, 28, 34, 56, 82, 89, 113, 154, 164, 244, 262, 488, 524, 730, 755, 802, 862, 1021, 1342, 1358, 1460, 2188, 2242, 2684, 2716, 3046, 4276, 4376, 4484, 6562, 6641, 6778, 8030, 8215, 8350, 8887, 12482, 13124, 14810, 19684

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..400 (first 100 terms from Harvey P. Dale)
Patrick De Geest, Palindromic Squares in bases 2 to 17
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, A007089, A029985, A263607, A263608.

pal3Q[n_]:=Module[{idn3=IntegerDigits[n^2,3]},idn3==Reverse[idn3]]; Select[Range[0,20000],pal3Q]  (* Harvey P. Dale, May 22 2012 *)

A029983 Squares which are palindromes in base 2.

Original entry on oeis.org

0, 1, 9, 20457529, 143784081, 331130809, 20074072489, 1193532215121, 10036851273801, 41413201925481, 155991531977649, 320642706437001, 4665141483989281, 87463589042698969, 152191954834044129

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..22
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A003166.

from itertools import count, islice
def A029983_gen(): # generator of terms
    return filter(lambda k: (s:=bin(k)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],(k**2 for k in count(0)))
A029983_list = list(islice(A029983_gen(),10)) # Chai Wah Wu, Jun 23 2022

A029989 Squares which are palindromes in base 5.

Original entry on oeis.org

0, 1, 4, 36, 676, 961, 4356, 15876, 24336, 391876, 423801, 571536, 646416, 9771876, 10732176, 14107536, 81974916, 244171876, 248094001, 264908176, 339812356, 351787536, 1061326084, 1167042244, 2181263616, 6103671876

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..147
Patrick De Geest, Palindromic Squares in bases 2 to 17
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, A263611, A263612, A029988.

pal5Q[n_]:=Module[{idn5=IntegerDigits[n,5]},idn5==Reverse[idn5]]; Select[ Range[0,80000]^2,pal5Q] (* Harvey P. Dale, May 05 2012 *)

A029985 Squares which are palindromes in base 3.

Original entry on oeis.org

0, 1, 4, 16, 100, 121, 400, 484, 784, 1156, 3136, 6724, 7921, 12769, 23716, 26896, 59536, 68644, 238144, 274576, 532900, 570025, 643204, 743044, 1042441, 1800964, 1844164, 2131600, 4787344, 5026564, 7203856, 7376656, 9278116

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..521
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A029984, A263607, A263608; also A002778, A003166.

b3pQ[n_]:=Module[{idn3=IntegerDigits[n,3]},idn3==Reverse[idn3]]; Select[ Range[0,3200]^2,b3pQ] (* Harvey P. Dale, Aug 07 2011 *)

A029986 Numbers k such that k^2 is palindromic in base 4.

Original entry on oeis.org

0, 1, 5, 17, 21, 65, 71, 83, 257, 273, 281, 317, 1025, 1055, 4097, 4161, 4193, 4401, 5157, 5179, 5221, 16385, 16511, 16865, 17239, 65537, 65793, 65921, 66753, 68695, 69521, 69777, 80739, 82053, 82171, 82309, 82885, 83301, 262145

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. A007090, A014192.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), this sequence (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), A030073 (b=15), A029733 (b=16), A118651 (b=17).

Select[Range[0,300000],IntegerDigits[#^2,4]==Reverse[ IntegerDigits[ #^2,4]]&] (* Harvey P. Dale, Dec 01 2015 *)

isok(k) = my(d=digits(k^2,4)); d == Vecrev(d); \\ Michel Marcus, Jul 04 2021

A029988 Numbers k such that k^2 is palindromic in base 5.

Original entry on oeis.org

0, 1, 2, 6, 26, 31, 66, 126, 156, 626, 651, 756, 804, 3126, 3276, 3756, 9054, 15626, 15751, 16276, 18434, 18756, 32578, 34162, 46704, 78126, 78876, 81276, 93756, 390626, 391251, 393876, 406276, 468756, 487981, 1166454, 1953126, 1956876

Offset: 1

Patrick De Geest

Patrick De Geest, Palindromic Squares in bases 2 to 17
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, A007091, A263611, A263612, A029989.

pal5Q[n_]:=Module[{idn5=IntegerDigits[n^2,5]},idn5==Reverse[idn5]]; Select[ Range[ 0,2*10^6],pal5Q] (* Harvey P. Dale, Feb 02 2023 *)

A050250 Number of nonzero palindromes less than 10^n.

Original entry on oeis.org

9, 18, 108, 198, 1098, 1998, 10998, 19998, 109998, 199998, 1099998, 1999998, 10999998, 19999998, 109999998, 199999998, 1099999998, 1999999998, 10999999998, 19999999998, 109999999998, 199999999998, 1099999999998, 1999999999998, 10999999999998

Offset: 1

Eric W. Weisstein, Dec 11 1999

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dr. Math, Palindromic Numbers.
Dr. Math, Palindromic Numbers.
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Palindromic Number.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A002113, A002778, A050683.

A050250List := proc(len);  local s, egf, ser; s:= 11/(2*sqrt(10));
egf := -2*exp(x) + (1-s)*exp(-sqrt(10)*x) + (1+s)*exp(sqrt(10)*x);
ser := series(egf, x, len+2): seq(simplify(n!*coeff(ser,x,n)), n = 1..len) end:
A050250List(25); # Peter Luschny, Jun 11 2022 after Stefano Spezia

LinearRecurrence[{1,10,-10},{9,18,108},30] (* Harvey P. Dale, Jan 29 2012 *)
CoefficientList[Series[2Cosh[Sqrt[10]x]-2(Cosh[x]+Sinh[x])+11Sinh[Sqrt[10]x]/Sqrt[10],{x,0,25}],x]Table[n!,{n,0,25}] (* Stefano Spezia, Jun 11 2022 *)

a(n)=10^(n\2)*(13-9*(-1)^n)/2-2 \\ Charles R Greathouse IV, Jun 25 2017

def a(n):
  m = 10 ** (n >> 1)
  if n & 1 == 0:
    return (m - 1) << 1
  else:
    return (11 * m) - 2 # Darío Clavijo, Oct 16 2023

a(2*k) = 2*10^k - 2, a(2*k + 1) = 11*10^k - 2. - Sascha Kurz, Apr 14 2002
From Jonathan Vos Post, Jun 18 2008: (Start)
a(n) = Sum_{i=1..n} A050683(i).
a(n) = Sum_{i=1..n} 9*10^floor((i-1)/2).
a(n) = 9*Sum_{i=1..n} 10^floor((i-1)/2). (End)
From Bruno Berselli, Feb 15 2011: (Start)
G.f.: 9*x*(1+x)/((1-x)*(1-10*x^2)).
a(n) = (1/2)*10^((2*n + (-1)^n - 1)/4)*(13 - 9*(-1)^n) - 2. (End)
a(1)=9, a(2)=18, a(3)=108; for n>3, a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3). - Harvey P. Dale, Jan 29 2012
a(n) = 10*a(n-2) + 18. - R. J. Mathar, Nov 07 2015
E.g.f.: 2*cosh(sqrt(10)*x) - 2*(cosh(x) + sinh(x)) + 11*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Jun 11 2022

More terms from Patrick De Geest, Dec 15 1999

a(24)-a(25) from Jonathan Vos Post, Jun 18 2008

cp's OEIS Frontend

A175440 Terms in A177950 that are not in A002778.

Views

Author

Keywords

Comments

Links

Crossrefs

A002779 Palindromic squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

Scala

Formula

A003166 Numbers whose square in base 2 is a palindrome.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A029984 Numbers k such that k^2 is palindromic in base 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A029983 Squares which are palindromes in base 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A029989 Squares which are palindromes in base 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A029985 Squares which are palindromes in base 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A029986 Numbers k such that k^2 is palindromic in base 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI