A027722 -id:A027722 - cp's OEIS frontend

A027723 Palindromes of form k^2 + k + 7.

7, 9, 313, 999, 31513, 75357, 78687, 90909, 98289, 3159513, 7642467, 9009009, 743080347, 900090009, 31413131413, 90000900009, 97474147479, 3105075705013, 9000009000009, 757082131280757, 900000090000009, 907340818043709, 90000000900000009, 92269201110296229

Offset: 1

Patrick De Geest

From Robert Israel, May 16 2018: (Start)
Palindromes m such that 4*m - 27 is a square.
Each term has an odd number of digits and ends in 3, 7 or 9.
Contains 9*(1+10^k+10^(2*k)) for each k>=1. (End)

Giovanni Resta, Table of n, a(n) for n = 1..45
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027722, A027692, A027756, A005471, A027721, A027725.

R[1]:= [1,3,5,7,9]: X[1]:= R[1]:
for k from 2 to 6 do
  R[k]:= map(t -> seq(10^(k-1)*j+t,j=0..9),R[k-1]);
X[k]:= map(t -> seq(j+10*t,j=0..9),X[k-1])
od:
Res:= 7,9:
for k from 1 to 6 do
  for j from 1 to 5*10^(k-1) do
      r:= 10^(k+1)*X[k][j]+R[k][j];
      for y from 0 to 9 do
        if issqr(4*(r+10^k*y)-27) then
          x:= r+10^k*y;
          Res:= Res,x;
        fi
od od od:
Res; # Robert Israel, May 16 2018

More terms from Giovanni Resta, Aug 28 2018

A027724 Numbers k such that k^2+k+8 is a palindrome.

Original entry on oeis.org

0, 29, 202, 285, 2925, 2935, 20377, 29570, 297119, 2834699, 2837875, 2990390, 2997334, 287010920, 2926428849, 202542945597, 295431039629, 21495814697072, 21614586653852

Offset: 1

Patrick De Geest

From Matthew L. LaSelle, Feb 23 2025: (Start)
k^2+k+8 only ends in a 0, 4, or 8; thus, if it is a palindrome, it only begins and ends in a 4 or 8.
a(20) > 2.174 * 10^13. (End)

Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027725, A027693, A027722, A027726.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 8; Select[Range[0, 3*10^5], palQ@ f@ # &] (* Giovanni Resta, Aug 29 2018 *)

isok(k) = my(d=digits(k^2+k+8)); d == Vecrev(d); \\ Michel Marcus, Feb 24 2025

a(14)-a(19) from Giovanni Resta, Aug 29 2018

A027729 Numbers k such that k^2+k+6 is a palindrome.

Original entry on oeis.org

0, 1, 24, 25, 28, 288, 2485, 2550, 2888, 2946, 28888, 146777, 264334, 288888, 292276, 2834101, 2873233, 2888888, 2952316, 2960816, 2985943, 2995631, 14604657, 16353547, 28888888, 29190748, 29585508, 148278137, 264056434, 288888888, 2853889203, 2931604151, 28988127118

Offset: 1

Patrick De Geest

Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027721, A027691, A027718, A027722.

[n: n in [0..3*10^6] | Intseq(n^2+n+6) eq Reverse(Intseq(n^2+n+6))]; // Vincenzo Librandi, Jun 16 2016

palOblongPlus6Q[n_] := Module[{d = IntegerDigits[n^2 + n + 6]}, d == Reverse[d]]; Select[Range[0, 3000000],  palOblongPlus6Q] (* Harvey P. Dale, Nov 14 2012 *)

a(23)-a(33) from Giovanni Resta, Aug 27 2018

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A027723 Palindromes of form k^2 + k + 7.

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A027724 Numbers k such that k^2+k+8 is a palindrome.

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A027729 Numbers k such that k^2+k+6 is a palindrome.

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Mathematica

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