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# 28

Please do not rely on any information it contains.

28 is the second perfect number, the first to not be squarefree.

## Membership in core sequences

 Even numbers ..., 22, 24, 26, 28, 30, 32, 34, ... A005843(14) Composite numbers ..., 25, 26, 27, 28, 30, 32, 33, ... A002808 Perfect numbers 6, 28, 496, 8128, 33550336, ... A000396 Triangular numbers ..., 10, 15, 21, 28, 36, 45, 55, ... A000217 Loeschian numbers ..., 21, 25, 27, 28, 31, 36, 37, ... A003136

In Pascal's triangle, 28 occurs four times, the first two times on the eighth row as the sum of 7 and 21. In Lozanić's triangle, 28 also first appears in the eighth row, though as the sum of 12, 19 and a tacit −3 from the previous row.

## Sequences pertaining to 28

 Divisors of 28 1, 2, 4, 7, 14, 28 A018254 Multiples of 28 0, 28, 56, 84, 112, 140, 168, 196, ... A135628 28-gonal numbers 1, 28, 81, 160, 265, 396, 553, 736, ... A161935 Centered 28-gonal numbers 1, 29, 85, 169, 281, 421, 589, 785, ... A195314 28-gonal pyramidal numbers 1, 29, 110, 270, 535, 931, 1484, ... A256648 $3x+1$ sequence starting at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, ... A033479 $3x-1$ sequence starting at 36 936, 18, 9, 26, 13, 38, 19, 56, 28, ... A008894 $5x+1$ sequence starting at 11 11, 56, 28, 14, 7, 36, 18, 9, 46, 23, ... A259193

## Partitions of 28

There are 3718 partitions of 28. There are several nontrivial partitions of 28 into its own divisors, but since it is a perfect number, exactly only one of those consists of distinct divisors.

There are only two Goldbach representations of 28: 5 + 23 and 11 + 17.

## Roots and powers of 28

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

 ${\sqrt {28}}$ 5.29150262 A010483 28 2 784 ${\sqrt[{3}]{28}}$ 3.03658897 A010599 28 3 21952 ${\sqrt[{4}]{28}}$ 2.30032663 A011023 28 4 614656 ${\sqrt[{5}]{28}}$ 1.94729436 A011113 28 5 17210368 ${\sqrt[{6}]{28}}$ 1.74258112 28 6 481890304 ${\sqrt[{7}]{28}}$ 1.60967004 28 7 13492928512 ${\sqrt[{8}]{28}}$ 1.51668277 28 8 377801998336 ${\sqrt[{9}]{28}}$ 1.44808927 28 9 10578455953408 ${\sqrt[{10}]{28}}$ 1.39545489 28 10 296196766695424 A009972

## Logarithms and 28th powers

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

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

 $\log _{28}2$ 0.208015 $\log _{2}28$ 4.80735 2 28 2.68435e+08 $\log _{28}e$ 0.300102 $\log 28$ 3.3322 A016651 $e^{28}$ 1.44626e+12 $\log _{28}3$ 0.329695 $\log _{3}28$ 3.0331 3 28 2.28768e+13 $\log _{28}\pi$ 0.343535 $\log _{\pi }28$ 2.91091 $\pi ^{28}$ 8.3214e+13 $\log _{28}4$ 0.416029 $\log _{4}28$ 2.40368 4 28 7.20576e+16 $\log _{28}5$ 0.482995 $\log _{5}28$ 2.07042 5 28 3.72529e+19 $\log _{28}6$ 0.53771 $\log _{6}28$ 1.85974 6 28 6.14094e+21 $\log _{28}7$ 0.583971 $\log _{7}28$ 1.71241 7 28 4.59987e+23 $\log _{28}8$ 0.624044 $\log _{8}28$ 1.60245 8 28 1.93428e+25 $\log _{28}9$ 0.659391 $\log _{9}28$ 1.51655 9 28 5.23348e+26 $\log _{28}10$ 0.69101 $\log _{10}28$ 1.44716 10 28 1e+28

See A122969 for the 28th powers of integers.

## Values for number theoretic functions with 28 as an argument

 $\mu (28)$ 0 $M(28)$ –1 $\pi (28)$ 9 $\sigma _{1}(28)$ 56 Note that this is twice 28. $\sigma _{0}(28)$ 6 $\phi (28)$ 12 $\Omega (28)$ 3 $\omega (28)$ 2 $\lambda (28)$ 6 This is the Carmichael lambda function. $\lambda (28)$ –1 This is the Liouville lambda function. $\zeta (28)$ 1.0000000037253340247884570548192... 28! 304888344611713860501504000000 $\Gamma (28)$ 10888869450418352160768000000

## Factorization of 28 in some quadratic integer rings

As was mentioned above, the prime factorization of 28 is $2^{2}\times 7$ in $\mathbb {Z}$ . But it has different factorizations in some quadratic integer rings.

 $\mathbb {Z} [i]$ $(1-i)^{2}(1+i)^{2}7$ $\mathbb {Z} [{\sqrt {-2}}]$ $({\sqrt {-2}})^{4}7$ $\mathbb {Z} [{\sqrt {2}}]$ $({\sqrt {2}})^{4}(3\pm {\sqrt {2}})$ $\mathbb {Z} [\omega ]$ $2^{2}(-2+\omega )(-2+\omega ^{2})$ $\mathbb {Z} [{\sqrt {3}}]$ $(1\pm {\sqrt {3}})^{2}7$ $\mathbb {Z} [{\sqrt {-5}}]$ 2 2 × 7 $\mathbb {Z} [\phi ]$ 2 2 × 7 $\mathbb {Z} [{\sqrt {-6}}]$ $2^{2}(1\pm {\sqrt {-6}})$ $\mathbb {Z} [{\sqrt {6}}]$ $(2\pm {\sqrt {6}})^{2}7$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$ $(-1)\left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)^{2}({\sqrt {-7}})^{2}$ $\mathbb {Z} [{\sqrt {7}}]$ $(3\pm {\sqrt {7}})^{2}({\sqrt {7}})^{2}$ $\mathbb {Z} [{\sqrt {-10}}]$ 2 2 × 7 $\mathbb {Z} [{\sqrt {10}}]$ 2 2 × 7 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$ $\mathbb {Z} [{\sqrt {11}}]$ $(-1)(3\pm {\sqrt {11}})^{2}(2\pm {\sqrt {11}})$ $\mathbb {Z} [{\sqrt {-13}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$ 2 2 × 7 $\mathbb {Z} [{\sqrt {-14}}]$ $\mathbb {Z} [{\sqrt {14}}]$ $(-1)(4\pm {\sqrt {14}})^{2}(7\pm 2{\sqrt {14}})$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ $\mathbb {Z} [{\sqrt {15}}]$ 2 2 × 7 $\mathbb {Z} [{\sqrt {-17}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$ $\left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)^{2}7$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$ $2^{2}\left({\frac {3}{2}}\pm {\frac {\sqrt {-19}}{2}}\right)$ $\mathbb {Z} [{\sqrt {19}}]$ $(13\pm 3{\sqrt {19}})^{2}7$ ## Representation of 28 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 11100 1001 130 103 44 40 34 31 28 26 24 22 20 1D 1C 1B 1A 19 18

 $-1$ 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