Heptagonal number

A heptagonal number is a figurate number that represents a heptagon. The n-th heptagonal number is given by the formula

\frac{5n^2 - 3n}{2}

The first five heptagonal numbers.

The first few heptagonal numbers are:

1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, … (sequence A000566 in the OEIS)

Parity

The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number.

Generalized heptagonal numbers

A generalized heptagonal number is obtained by the formula

T_n + T_{\lfloor \frac{n}{2} \rfloor},

where T_n is the nth triangular number. The first few generalized heptagonal numbers are:

1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112, … (sequence A085787 in the OEIS)

Every other generalized heptagonal number is a regular heptagonal number. Besides 1 and 70, no generalized heptagonal numbers are also Pell numbers.^[1]

Sum of reciprocals

A formula for the sum of the reciprocals of the heptagonal numbers is given by:^[2]

\sum_{n=1}^\infty \frac{2}{n(5n-3)} = \frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\frac{2}{3}\ln(5)+\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)

Heptagonal roots

In analogy to the square root of x, one can calculate the heptagonal root of x, meaning the number of terms in the sequence up to and including x.

The heptagonal root of x is given by the formula

n={\frac {{\sqrt {40x+9}}+3}{10}}.

Derivation of heptagonal root formula

The heptagonal roots n of x are derived by:

x = \frac{5n^2 - 3n}{2}

2x = 5n^2 - 3n

5n^2 - 3n - 2x = 0

n = \frac{-(-3) \pm \sqrt{(-3)^2 - (4 \times 5 \times -2x)}}{2 \times 5}

(use quadratic formula to solve for n)

n = \frac{3 \pm \sqrt{9 - (-40x)}}{10}

n = \frac{3 \pm \sqrt{9 + 40x}}{10}

Rearrange this to:

n={\frac {\pm {\sqrt {40x+9}}+3}{10}},

and taking the only positive value gives the formula for n associated with a given x.

References

↑ B. Srinivasa Rao, "Heptagonal Numbers in the Pell Sequence and Diophantine equations $2x^2 = y^2(5y - 3)^2 \pm 2$ " Fib. Quart. 43 3: 194
↑ Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
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Recreational
mathematics

Base-dependent
numbers

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