Highly cototient number

In number theory, a branch of mathematics, a highly cototient number is a positive integer $k$ which is above 1 and has more solutions to the equation

x-\phi (x)=k

than any other integer below $k$ and above 1. Here, $\phi$ is Euler's totient function. There are infinitely many solutions to the equation for

k

= 1

so this value is excluded in the definition. The first few highly cototient numbers are:[1]

2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, ... (sequence A100827 in the OEIS)

Many of the highly cototient numbers are odd. In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 29 modulo 30.

The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Computations become harder, since integer factorization becomes harder as the numbers get larger.

Example

The cototient of $x$ is defined as $x-\phi (x)$ , i.e. the number of positive integers less than or equal to $x$ that have at least one prime factor in common with $x$ . For example, the cototient of 6 is 4 since these four positive integers have a prime factor in common with 6: 2, 3, 4, 6. The cototient of 8 is also 4, this time with these integers: 2, 4, 6, 8. There are exactly two numbers, 6 and 8, which have cototient 4. There are fewer numbers which have cototient 2 and cototient 3 (one number in each case), so 4 is a highly cototient number.

(sequence A063740 in the OEIS)

k (highly cototient k are bolded)	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
Number of solutions to x – φ(x) = k	1	∞	1	1	2	1	1	2	3	2	0	2	3	2	1	2	3	3	1	3	1	3	1	4	4	3	0	4	1	4	3

n	ks such that $k-\phi (k)=n$	number of ks such that $k-\phi (k)=n$ (sequence A063740 in the OEIS)
0	1	1
1	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ... (all primes)	∞
2	4	1
3	9	1
4	6, 8	2
5	25	1
6	10	1
7	15, 49	2
8	12, 14, 16	3
9	21, 27	2
10		0
11	35, 121	2
12	18, 20, 22	3
13	33, 169	2
14	26	1
15	39, 55	2
16	24, 28, 32	3
17	65, 77, 289	3
18	34	1
19	51, 91, 361	3
20	38	1
21	45, 57, 85	3
22	30	1
23	95, 119, 143, 529	4
24	36, 40, 44, 46	4
25	69, 125, 133	3
26		0
27	63, 81, 115, 187	4
28	52	1
29	161, 209, 221, 841	4
30	42, 50, 58	3
31	87, 247, 961	3
32	48, 56, 62, 64	4
33	93, 145, 253	3
34		0
35	75, 155, 203, 299, 323	5
36	54, 68	2
37	217, 1369	2
38	74	1
39	99, 111, 319, 391	4
40	76	1
41	185, 341, 377, 437, 1681	5
42	82	1
43	123, 259, 403, 1849	4
44	60, 86	2
45	117, 129, 205, 493	4
46	66, 70	2
47	215, 287, 407, 527, 551, 2209	6
48	72, 80, 88, 92, 94	5
49	141, 301, 343, 481, 589	5
50		0

Primes

The first few highly cototient numbers which are primes are [2]

2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889, 2099, 2309, 2729, 3359, 3989, 4289, 4409, 5879, 6089, 6719, 9029, 9239, ... (sequence A105440 in the OEIS)

References

Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation..
Sloane, N. J. A. (ed.). "Sequence A105440 (Highly cototient numbers that are prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Totient function
Euler's totient function $φ (n)$ Jordan's totient function $J k (n)$ Carmichael function (reduced totient function) $λ (n)$ Nontotient Noncototient Highly totient number Highly cototient number Sparsely totient number

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Weakly Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
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Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
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Highly cototient number

Example

Primes

See also

References