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Palindromic Heptagonals (or 7-gonals)
Factorization Records triangle square penta hexa octa nona

Palindromic numbers are numbers which read the same from

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

Here are a few random examples : 7, 3113, 44611644

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

base x ( 5 x base - 3 ) ------------------------------- 2

So far this compilation counts 85 Palindromic Heptagonals.

A palindromic heptagonal numbers can end with any digit.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.

There exist no palindromic heptagonals of length  8, 9, 14, 16, 32, 38, 44, 45, 46, 51, 52, 56.

(Sloane's A059869)

A B C →  10 7 1 

A B C →  10 17 7 

Sloane's A046194 gives the first numbers that are both Heptagonal and Triangular.

1, 55, 121771, 5720653, 12625478965, ...

Consult also Eric Weisstein's page Heptagonal Triangular Number.

Sloane's A036354 gives the first numbers that are both Heptagonal and Square.

1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, ...

Consult also Eric Weisstein's page Heptagonal Square Number.

Sloane's A048900 gives the first numbers that are both Heptagonal and Pentagonal.

1, 4347, 16701685, 246532939589097, ...

Consult also Eric Weisstein's page Heptagonal Pentagonal Number.

Sloane's A048903 gives the first numbers that are both Heptagonal and Hexagonal.

1, 121771, 12625478965, 1309034909945503, ...

Consult also Eric Weisstein's page Heptagonal Hexagonal Number.

Sloane's A048906 gives the first numbers that are both Heptagonal and Octagonal.

1, 297045, 69010153345, ...

Consult also Eric Weisstein's page Heptagonal Octagonal Number.

Sloane's A048921 gives the first numbers that are both Heptagonal and Nonagonal.

1, 26884, 542041975, 10928650279834, ...

Consult also Eric Weisstein's page Heptagonal Nonagonal Number.

The best way to get a 'structural' insight as how to imagine heptagonals is to visit for instance this site :

• From Eric Weisstein's Math Encyclopedia : Heptagonal Number

Interesting semiprime (two distinct primefactors of equal length)

Jeff Heleen (email submitted the first palindromic heptagonals [1] and [53].

Feng Yuan (email from Washington State, USA, discovered the palindromic heptagonals
starting from index number [54] and [55].

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, discovered the palindromic heptagonals
starting from index number [56] up to [71] alas with some missers.

[ November 4, 2022 ]
Robert Xiao (email) added two missing entries [57] and [62].

[ November 11, 2022 ]
Robert Xiao (email) added four missing entries [63], [66], [68] and [70].

[ November 11, 2022 ]
David Griffeath (email) using Rust code by Robert Xiao (email) added entry [74]. This was once a record holder.

[ December 6, 2022 ]
Robert Xiao (email) added two missing entries [72] and [73] and five new entries [75] up to [79]

[ January 11, 2023 ]
David Griffeath (email) using CUDA code by Robert Xiao (email) added two new entries [80] and [81].
An exhaustive search for 53-digit palindromic heptagonals was done. There are two, both with even roots.

[ June 15 & 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [82] up to [84].

[ September 25, 2024 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added one more entry [85].

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