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15

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15 is a composite number. In fact, it is the smallest composite number with the property that there is only one group of order (see A000001 and A050384).

Membership in core sequences

Odd numbers ..., 9, 11, 13, 15, 17, 19, 21, ... A005408
Composite numbers ..., 10, 12, 14, 15, 16, 18, ... A002808
Semiprimes 4, 6, 9, 10, 14, 15, 21, 22, ... A001358
Partition numbers 1, 1, 2, 3, 5, 7, 11, 15, 22, ... A000041
Triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, ... A000217

In Pascal's triangle, 15 occurs four times, the first two times surrounding the first occurrence of 20, with 6 on either side.

Sequences pertaining to 15

Multiples of 15 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, ... A008597
15-gonal numbers 1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870, ... A051867
sequence starting at 15 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, ... A033480

Partitions of 15

There are 176 partitions of 15.

The Goldbach representations of 15 using distinct primes are: 2 + 13 = 3 + 5 + 7.

Roots and powers of 15

In the table below, irrational numbers are given truncated to eight decimal places.

3.87298334 A010472 15 2 225
2.46621207 A010587 15 3 3375
1.96798967 A011012 15 4 50625
1.71877192 A011100 15 5 759375
1.57041780 A011350 15 6 11390625
1.47235670 A011351 15 7 170859375
1.40285055 A011352 15 8 2562890625
1.35106675 A011353 15 9 38443359375
1.31101942 A011354 15 10 576650390625
1.27913795 A011355 15 11 8649755859375
1.25316311 A011356 15 12 129746337890625
A001024

Logarithms and fifteenth powers

In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript.

As above, irrational numbers in the following table are truncated to eight decimal places.

TABLE

(See A000584 for the fifth powers of integers).

Values for number theoretic functions with 15 as an argument

1
–1
6
24
4
8
2
2
4 This is the Carmichael lambda function.
–1 This is the Liouville lambda function.
1.00003058823630702... (see A013673).
15! 1307674368000
87178291200

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −15, 15

Neither nor is a unique factorization domain. only has two units, which makes things simpler and gives us more confidence in identifying instances of multiple distinct factorizations. But there are still traps for the unwary, such as the presence of so-called "half-integers."

For example, we could say that 16 has three distinct factorizations: 2 4, and . But that would be wrong. Each number in the latter two factorizations has nontrivial non-unit divisors and thus probably don't correspond to distinct factorizations any more than 4 2 does. First notice that the "half-integer" is fully an algebraic integer, with minimal polynomial , and therefore an integer of this domain. Then verify that and . With adjustments of signs, the other "composite" factors are addressed.

This raises the question of whether is a distinct factorization. If it is distinct, it is rather inelegant.

As for , the of the MORE REMARKS GO HERE

2 Irreducible
3
4 2 2 OR 2 2
5 Irreducible
6 2 × 3 2 × 3 OR
7 Prime Irreducible
8 2 3
9 3 2
10 2 × 5 OR 2 × 5 OR
11 Prime
12 2 2 × 3
13 Prime
14 2 × 7 2 × 7 OR
15 3 × 5 OR 3 × 5 OR
16 2 4 OR 2 4
17 Irreducible
18 2 × 3 2
19 Prime
20 2 2 × 5

Ideals can help us make sense of the lack of unique factorization in these two rings.

Factorization of
In In
2
3
5
7 Prime
11 Prime
13 Prime
17
19 Prime
23
29
31
37
41
43
47

Factorization of 15 in some quadratic integer rings

In , 15 is composite, being the product of two primes. It is of course also composite in any quadratic integer rings, but it has different factorizations in some of those.

3 × 5
3 × 5 OR
3 × 5
3 × 5 OR
3 × 5
3 × 5 OR 3 × 5 OR
3 × 5 3 × 5

Representation of 15 in various bases

Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 through 36
Representation 1111 120 33 30 23 21 17 16 15 14 13 12 11 10 F

We see that 15 is a Harshad number in bases 3, 5, 6, 7, 11, 12, 13 and trivially in base 15.

See also

Some integers
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
1729

References