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Palindromic Quasi_Over_Squares of the form n^2+(n+1)
The Details n(n+0) n(n+1) n(n+2) n(n+x) n^2+1 n^2+x n^2–x n^2+(n+x)

  left to right (forwards) as from the right to left (backwards)

Quasi_Over_Square numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base2 + ( base + 1 )1

Let me name the numbers of the form n + (n+1)^2 or (n-1) + n^2 as follows : Quasi_Under_Squares.
Numbers of the form n^2 + (n+1) then become the Quasi_Over_Squares.
Readers with more original suggestions for naming these numbers can always send their proposals to me at my e-mail address.

26 (5445)_n 62, for n = 2 to 4
26545 (5445)_n 437, for n = 0 to 4
|Astonishing!| the largest known Quasi_Over_Square now has a length of 47 digits !

So far I compiled 129 Palindromic Quasi_Over_Squares.

All palindromic numbers of the form n^2 + (n+1) can only end with 1, 3 or 7

It is no coincidence that I didn't find Palindromic Quasi_Over_Squares of even length.

        n mod 11 :   0   1   2   3   4   5   6   7   8   9   10
(n^2+n+1) mod 11 :   1   3   7   2  10   9  10   2   7   3    1
[ if zero appeared in the second line then it would be divisible by 11 ]

Either n or n+1 is even, so their product n(n+1) is even.
Therefore n(n+1)+1 is odd !!
Simple isn't it.

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online : Neil Sloane's Integer Sequences One can find the regular numbers of the form n2+(n+1) at %N Central polygonal numbers: n^2 – n + 1 under A002061. The palindromic numbers of the form n2+(n+1) are categorised as follows : %N n^2 + (n+1) is a palindrome under A028413. %N Palindromes of the form n^2 + (n+1) under A028414. If one searches the database for the short sequence {1,3,7,13} you get a positive reply and one of the entries shown is : %N Primes of form n^2 + n + 1 under A002383. %N n^2 + n + 1 is prime under A002384. Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

n(n – 1) + 1 =

(m+1) ( (m+1) – 1) + 1 =

(m+1) (m) + 1 =

m^2 + m + 1 =

m^2 + (m+1) QED

A lot of the palindromes are semiprimes. Is there a logical explanation for this phenomenon ?

 [9]
F_qs  30.103 =
      PRIME

Comm  30103 is the only known multi-digit palindromic prime 
      of the form sigma(n)/d(n), when n is composite. ( McCranie and Honaker )
      sigma(n) = the sum of all the factors of n, including 1 and n itself.
      d(n) = the number of factors of n, including 1 and n itself.

      30103 is also nice because it comes up for
      n = 149645 (an odd number) and once for
      n = 179574 (an even number).

      n         sigma(n)  d(n)  sigma(n)/d(n)=palprime
      149645    180618    6     30103
      179574    361236    12    30103

      30103 is still the only "known" palprime with the property.

My program searched for palindromes exhaustively upto length 31.

From length 33 on these numbers were discovered by Feng Yuan during [ January 19-25, 2008 ].

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( © All rights reserved ) - Last modified : August 13, 2021.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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