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
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
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a002779 n = a002778_list !! (n-1)
a002779_list = filter ((== 1) . a136522) a000290_list
-- Reinhard Zumkeller, Oct 11 2011
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[k^2:k in [0..100000]| Intseq(k^2) eq Reverse(Intseq(k^2)) ]; // Marius A. Burtea, Oct 15 2019
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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 *)
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is(n)=my(d=digits(n)); d==Vecrev(d) && issquare(n) \\ Charles R Greathouse IV, Feb 06 2017
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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
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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
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
- 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.
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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
A354256
Squares that remain square when written backward, are not divisible by 10, and have an even number of digits.
Original entry on oeis.org
1089, 9801, 698896, 10036224, 42263001, 637832238736, 1021178969603881, 1883069698711201, 4099923883299904, 6916103777337773016196
Offset: 1
There are no 2-digit terms.
The smallest 4-digit multiple of 121 is 1089 = 33^2, which happens to be a(1); its digit reversal is a(2) = 9801 = 99^2.
The only 6-digit term is the palindrome a(3) = 698896 = 836^2.
The only 8-digit terms are a(4) = 10036224 = 3168^2 and its digit reversal a(5) = 42263001 = 6501^2.
There are no 10-digit terms.
The only 12-digit term is the palindrome a(6) = 637832238736 = 798644^2.
There are no 14-digit terms.
There are three 16-digit terms: a(7) = 1021178969603881 = 31955891^2, its digit reversal a(8) = 1883069698711201 = 43394351^2, and the palindrome a(9) = 4099923883299904 = 64030648^2.
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Select[Range[500000]^2,EvenQ[IntegerLength[#]]&&Mod[#,10]!=0&&IntegerQ[Sqrt[ IntegerReverse[ #]]]&] (* The program generates the first five terms of the sequence. *) (* Harvey P. Dale, Jul 28 2024 *)
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from math import isqrt
from itertools import count, islice
def sqr(n): return isqrt(n)**2 == n
def agen(): yield from (k*k for k in count(1) if k%10 and len(s:=str(k*k))%2==0 and sqr(int(s[::-1])))
print(list(islice(agen(), 6))) # Michael S. Branicky, May 23 2022
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from math import isqrt
from itertools import count, islice
from sympy import integer_nthroot
def A354256_gen(): # generator of terms
for l in count(2,2):
for m in (1,4,5,6,9):
for k in range(1+isqrt(m*10**(l-1)-1),1+isqrt((m+1)*10**(l-1)-1)):
if k % 10 and integer_nthroot(int(str(k*k)[::-1]),2)[1]:
yield k*k
A354256_list = list(islice(A354256_gen(),9)) # Chai Wah Wu, May 23 2022
Showing 1-3 of 3 results.
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