A002778 -id:A002778 - cp's OEIS frontend

A029990 Numbers k such that k^2 is palindromic in base 6.

0, 1, 2, 7, 37, 43, 76, 91, 217, 259, 1064, 1297, 1333, 1519, 1555, 2704, 3367, 7777, 8029, 9079, 19747, 46657, 46873, 47989, 48205, 54439, 54655, 54695, 83979, 118027, 241304, 279937, 281449, 287749, 326599, 707707, 1679617

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..92
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A007092.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), this sequence (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).

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Select[Range[1000], palindromicQ[#^2, 6] &] (* Alonso del Arte, Mar 05 2017 *)

ispal(n,base)=my(d=digits(n,base)); d==Vecrev(d)
is(n)==ispal(n^2,6) \\ Charles R Greathouse IV, Mar 09 2017

A029992 Numbers k such that k^2 is palindromic in base 7.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 11, 20, 32, 40, 50, 57, 64, 80, 160, 200, 344, 400, 500, 550, 557, 730, 1000, 1376, 1432, 1892, 2402, 2451, 2500, 2752, 2801, 3440, 3784, 3902, 5101, 5266, 6880, 8296, 9460, 9608, 9804, 16808, 17200, 19216, 19608, 22693

Offset: 1

Patrick De Geest

			8^2 = 64, which is 121 in base 7, and since that's palindromic, 8 is in the sequence.
9^2 = 81, which is 144 in base 7, but since that's not palindromic, 9 is not in the sequence.

Marius A. Burtea, Table of n, a(n) for n = 1..237
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002440 (squares written in base 7), A007093.

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), this sequence (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).

[k:k in [0..23000]| Seqint(Intseq(k^2,7)) eq Seqint(Reverse(Intseq(k^2,7)))]; // Marius A. Burtea, Jan 22 2020

Select[Range[0, 16806], IntegerDigits[#^2, 7] == Reverse[IntegerDigits[#^2, 7]] &] (* Alonso del Arte, Jan 21 2020 *)

(0 to 16806).filter(n => Integer.toString(n * n, 7) == Integer.toString(n * n, 7).reverse) // Alonso del Arte, Jan 21 2020

A029733 Numbers k such that k^2 is palindromic in base 16.

Original entry on oeis.org

0, 1, 2, 3, 17, 34, 257, 273, 289, 305, 319, 514, 530, 546, 773, 1377, 4097, 4369, 4641, 8194, 8254, 8466, 8734, 9046, 51629, 65537, 65793, 66049, 66305, 69649, 69905, 70161, 70417, 73505, 73761, 74017, 74273, 76879, 92327, 131074

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), A030073 (b=15), this sequence (b=16), A118651 (b=17).

n2palQ[n_]:=Module[{id=IntegerDigits[n^2,16]},id==Reverse[id]]; Select[ Range[ 0,150000],n2palQ] (* Harvey P. Dale, Mar 31 2018 *)

from itertools import count, islice
def A029733_gen(): # generator of terms
    return filter(lambda k: (s:=hex(k**2)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A029733_list = list(islice(A029733_gen(),20)) # Chai Wah Wu, Jun 23 2022

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

Original entry on oeis.org

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).

A034822 Numbers k such that there are no palindromic squares of length k.

Original entry on oeis.org

2, 4, 8, 10, 14, 18, 20, 24, 30, 38, 40

Offset: 1

Patrick De Geest, Oct 15 1998

All terms are even since (10^k+1)^2 is a palindrome of length 2*k+1. a(12) >= 46 if it exists (see A263618). - Chai Wah Wu, Jun 14 2024

Patrick De Geest, Palindromic Squares in bases 2 to 17
Eric Weisstein's World of Mathematics, Palindromic Number

Cf. A002778, A002779, A034307, A263618.

A034822[n_] := Select[Range[Ceiling[Sqrt[10^(n - 1)]], Floor[Sqrt[10^n]]], #^2 == IntegerReverse[#^2] &];
Select[Range[12], Length[A034822[#]] == 0 &] (* Robert Price, Apr 23 2019 *)

from sympy import integer_nthroot as iroot
def ispal(n): s = str(n); return s == s[::-1]
def ok(n):
  for r in range(iroot(10**(n-1), 2)[0] + 1, iroot(10**n, 2)[0]):
    if ispal(r*r): return False
  return True
print([m for m in range(1, 16) if ok(m)]) # Michael S. Branicky, Feb 04 2021

Two more terms from Patrick De Geest, Apr 01 2002

A118651 Numbers k such that k^2 is a palindrome when written in base 17.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 12, 18, 28, 36, 84, 108, 290, 307, 324, 341, 580, 597, 614, 1080, 1614, 1740, 1842, 2616, 3378, 3480, 3720, 4344, 4824, 4914, 5220, 5526, 6408, 9828, 10134, 10440, 14472, 17944, 19336, 24360, 27624, 29484, 31320, 33144, 33960

Offset: 1

Neven Juric (neven.juric(AT)apis-it.hr), May 12 2006

			E.g. 4^2 = 16_10 = G_16, 6^2 = 36_10 = 22_17, etc.

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

Cf. A029984 for base 3, A029986 for base 4, A029988 for base 5, A029990 for base 6, A029992 for base 7, A029805 for base 8, A029994 for base 9, A002778 for base 10, A029996 for base 11, A029733 for base 16

Cf. also A016106, A028818, A059744, A059745.

A002780 Numbers whose cube is a palindrome.

Original entry on oeis.org

0, 1, 2, 7, 11, 101, 111, 1001, 2201, 10001, 10101, 11011, 100001, 101101, 110011, 1000001, 1001001, 1100011, 10000001, 10011001, 10100101, 11000011, 100000001, 100010001, 100101001, 101000101, 110000011, 1000000001, 1000110001

Offset: 1

N. J. A. Sloane

a(8)=2201 is the only known non-palindromic rootnumber.
There are no further non-palindromic terms (other than 2201) up to 10^11. - Matevz Markovic, Apr 04 2011. There are none up to 10^15, by direct search. - Charles R Greathouse IV, May 16 2011
There are no non-palindromic terms in the range 10^15 to 10^20 with digits from the set {0,1,2}. - Hans Havermann, May 18 2011.
From Vladimir Shevelev, May 23 2011: (Start)
Using the table by Noe-De Geest, I noticed that all numbers {a(n)=A002780(n);  11<=a(n)<=10^17+10^16+11}, except 2201, allow a partition into 3 disjoint classes of terms of the following forms: 10^k+1, 10^(2*k)+10^k+1, and (10^u+1)*(10^v+1).
Does there exist a term a(n)>10^17+10^16+11 which is in none of these classes?
If there is no such term, then we conclude that the sum of digits of a(n) does not exceed 4 (more exactly, it is i+1 where i is the number of class).
One can prove that the sequence contains no term (other than 2201) with sum of digits = 5. (End)

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).

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]

Cf. A002781 (cubes of these numbers).

isok(k) = my(d=digits(k^3)); Vecrev(d) == d; \\ Michel Marcus, Aug 02 2022

def ispal(s): return s == s[::-1]
def ok(n): return ispal(str(n**3))
print([k for k in range(10**7) if ok(k)]) # Michael S. Branicky, Aug 02 2022

More terms from Patrick De Geest

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 *)

cp's OEIS Frontend

A029990 Numbers k such that k^2 is palindromic in base 6.

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A029992 Numbers k such that k^2 is palindromic in base 7.

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A029733 Numbers k such that k^2 is palindromic in base 16.

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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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A034822 Numbers k such that there are no palindromic squares of length k.

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A118651 Numbers k such that k^2 is a palindrome when written in base 17.

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A002780 Numbers whose cube is a palindrome.

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