A027720 -id:A027720 - cp's OEIS frontend

A070253 Numbers k such that k^2 - 1 is a palindrome.

1, 2, 3, 10, 18, 24, 65, 76, 100, 192, 205, 1000, 1748, 1908, 2366, 2967, 5732, 10000, 18992, 20565, 100000, 174602, 174748, 179318, 243064, 293787, 552102, 1000000, 1868288, 2967033, 9200157, 10000000, 22765896, 31552660, 93809717, 100000000

Offset: 1

Amarnath Murthy, May 06 2002

Every palindrome of the form h^2-1 is of the form m*(m+2) (easy to prove by replacing h by m+1). In fact this is equal to A028503 + 1. - Patrick De Geest, May 09 2002

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

Cf. A027719, A027720, A070254, A028503.

Do[ If[ a = IntegerDigits[n^2 - 1]; a == Reverse[a], Print[n]], {n, 1, 10^8/4}]
Select[Range[10^8],PalindromeQ[#^2-1]&] (* Harvey P. Dale, Oct 13 2024 *)

intreverse(n)=local(d,rev); rev=0; while(n>0,d=divrem(n,10); n=d[1]; rev=10*rev+d[2]); rev
for(n=1,100000000,q=n*n-1; if(q==intreverse(q),print1(n,",")))

a(n) = A028503(n) + 1. - Giovanni Resta, Aug 29 2018

Edited by Jason Earls, Klaus Brockhaus and Robert G. Wilson v, May 08 2002

A027719 Numbers k such that k^2 + 1 is a palindrome.

Original entry on oeis.org

0, 1, 2, 10, 25, 100, 1000, 1020, 1489, 2248, 10000, 10090, 100000, 100910, 102020, 167491, 1000000, 1000200, 1009090, 2744934, 10000000, 10000900, 10090910, 24917195, 100000000, 100909090, 103226660, 271867456, 1000000000, 1000002000, 1009090910, 1577033471

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..48
P. De Geest, Palindromic incremented squares of the form n^2+1

Cf. A002522, A027720, A002778, A070253.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; Select[Range[0, 10^5], palQ[#^2 + 1] &] (* Giovanni Resta, Aug 29 2018 *)

More terms from Giovanni Resta, Aug 28 2018

A070254 Perfect squares one more than a palindrome.

Original entry on oeis.org

1, 4, 9, 100, 324, 576, 4225, 5776, 10000, 36864, 42025, 1000000, 3055504, 3640464, 5597956, 8803089, 32855824, 100000000, 360696064, 422919225, 10000000000, 30485858404, 30536863504, 32154945124, 59080108096, 86310801369, 304816618404, 1000000000000, 3490500050944

Offset: 1

Amarnath Murthy, May 06 2002

All even powers of 10 are members of both A070254 and A027720.

Giovanni Resta, Table of n, a(n) for n = 1..53

Cf. A028504, A027719, A027720, A070253.

Do[ If[a = IntegerDigits[n^2 - 1]; a == Reverse[a], Print[n^2]], {n, 1, 10^6}]
Select[Range[300000]^2,PalindromeQ[#-1]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 06 2018 *)

a(n) = A070253(n)^2 = A028504(n) + 1. - Giovanni Resta, Aug 29 2018

Edited by Jason Earls and Robert G. Wilson v, May 08 2002

Offset changed by and more terms from Giovanni Resta, Aug 28 2018

A124664 Both k and its reverse are one more than a square.

Original entry on oeis.org

1, 2, 5, 10, 50, 101, 626, 730, 1090, 2210, 5477, 7745, 10001, 10610, 71290, 227530, 1000001, 1010026, 1014050, 1040401, 2217122, 2676497, 5053505, 5631130, 6200101, 6265010, 7946762, 100000001, 101808101, 248157010, 10000000001, 10180608202, 10182828101

Offset: 1

Tanya Khovanova, Dec 23 2006

The first digit for each term is either 1, 2, 5, 6 or 7. - Chai Wah Wu, May 25 2017

			5477 is in the sequence because 5477 = 74^2 + 1 and 7745 = 88^2 + 1.

Chai Wah Wu, Table of n, a(n) for n = 1..100
Erich Friedman, What's Special About This Number?. See 5477 and 7745.

A066618 is a subsequence of this sequence of numbers that do not end in 0. The sequence A027720 = Palindromes of form n^2 + 1 - is a palindromic subsequence of this sequence.

Cf. A287389: both k and its reverse are one less than a square.

r:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
select(x-> issqr(r(x)-1), [n^2+1$n=0..150000])[]; # Alois P. Heinz, May 24 2017

Select[Range[10000000], IntegerQ[Sqrt[ # - 1]] && IntegerQ[Sqrt[FromDigits[Reverse[IntegerDigits[ # ]]] - 1]] &]

More terms from Alois P. Heinz, May 24 2017

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A070253 Numbers k such that k^2 - 1 is a palindrome.

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A027719 Numbers k such that k^2 + 1 is a palindrome.

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A070254 Perfect squares one more than a palindrome.

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A124664 Both k and its reverse are one more than a square.

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