A centered heptagonal number is a
centered figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
* polygon ...
that represents a
heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.
The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived num ...
with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The centered heptagonal number for ''n'' is given by the formula
:
.
The first few centered heptagonal numbers are
1,
8,
22,
43,
71,
106,
148 148 may refer to:
*148 (number), a natural number
*AD 148, a year in the 2nd century AD
*148 BC, a year in the 2nd century BC
*148 (album), an album by C418
*148 (Meiktila) Battery Royal Artillery
*148 (New Jersey bus) 148 may refer to:
*148 (numb ...
,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
, 253, 316, 386, 463, 547, 638, 736, 841, 953
Properties
* Centered heptagonal numbers alternate parity in the pattern odd-even-even-odd.
* A heptagonal numbers can expressed as a multiple of a
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
by 7, plus one:
:
*
is the sum of the integers between n+1 and 3n+1 (including) minus the sum of the integers from 0 to n (including).
Centered heptagonal prime
A centered heptagonal prime is a centered heptagonal number that is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. The first few centered heptagonal primes are
:43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, ...
Due to parity, the centered heptagonal primes are in the form of
or
.
The centered heptagonal
twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
numbers are
:43, 71, 197, 463, 1933, 5741, 8233, 9283, 11173, 14561, 34651, ... .
See also
*Regular
heptagonal number A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The ''n''-th heptagonal number is given by the formula
:H_n=\frac.
The first few heptagonal numbers are:
: 0, 1, 7, 18, 34, 55, 81, 112, ...
.
{{Classes of natural numbers
Figurate numbers