Noncototient

In mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototient is a number that is never a cototient.

It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number n can be represented as a sum of two distinct primes p and q, then

pq-\varphi (pq)=pq-(p-1)(q-1)=p+q-1=n-1.\,

It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations $1=2-\phi (2),3=9-\phi (9)$ and $5=25-\phi (25)$ .

For even numbers, it can be shown

2pq-\varphi (2pq)=2pq-(p-1)(q-1)=pq+p+q-1=(p+1)(q+1)-2

Thus, all even numbers n such that n+2 can be written as (p+1)*(q+1) with p, q primes are cototients.

The first few noncototients are

10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, 206, 218, 222, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, ... (sequence A005278 in the OEIS)

The cototient of n are

0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 46, 1, 48, ... (sequence A051953 in the OEIS)

Least k such that the cototient of k is n are (start with n = 0, 0 if no such k exists)

1, 2, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36, 69, 0, 63, 52, 161, 42, 87, 48, 93, 0, 75, 54, 217, 74, 99, 76, 185, 82, 123, 60, 117, 66, 215, 72, 141, 0, ... (sequence A063507 in the OEIS)

Greatest k such that the cototient of k is n are (start with n = 0, 0 if no such k exists)

1, ∞, 4, 9, 8, 25, 10, 49, 16, 27, 0, 121, 22, 169, 26, 55, 32, 289, 34, 361, 38, 85, 30, 529, 46, 133, 0, 187, 52, 841, 58, 961, 64, 253, 0, 323, 68, 1369, 74, 391, 76, 1681, 82, 1849, 86, 493, 70, 2209, 94, 589, 0, ... (sequence A063748 in the OEIS)

Number of ks such that k-φ(k) is n are (start with n = 0)

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, 3, 4, 3, 0, 5, 2, 2, 1, 4, 1, 5, 1, 4, 2, 4, 2, 6, 5, 5, 0, 3, 0, 6, 2, 4, 2, 5, 0, 7, 4, 3, 1, 8, 4, 6, 1, 3, 1, 5, 2, 7, 3, ... (sequence A063740 in the OEIS)

Erdős (1913-1996) and Sierpinski (1882-1969) asked whether there exist infinitely many noncototients. This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family $2^{k}\cdot 509203$ is an example (See Riesel number). Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca (2000).

n	numbers k such that k-φ(k) = n	n	numbers k such that k-φ(k) = n	n	numbers k such that k-φ(k) = n	n	numbers k such that k-φ(k) = n
1	all primes	37	217, 1369	73	213, 469, 793, 1333, 5329	109	321, 721, 1261, 2449, 2701, 2881, 11881
2	4	38	74	74	146	110	150, 182, 218
3	9	39	99, 111, 319, 391	75	207, 219, 275, 355, 1003, 1219, 1363	111	231, 327, 535, 1111, 2047, 2407, 2911, 3127
4	6, 8	40	76	76	148	112	196, 208
5	25	41	185, 341, 377, 437, 1681	77	245, 365, 497, 737, 1037, 1121, 1457, 1517	113	545, 749, 1133, 1313, 1649, 2573, 2993, 3053, 3149, 3233, 12769
6	10	42	82	78	114	114	226
7	15, 49	43	123, 259, 403, 1849	79	511, 871, 1159, 1591, 6241	115	339, 475, 763, 1339, 1843, 2923, 3139
8	12, 14, 16	44	60, 86	80	152, 158	116
9	21, 27	45	117, 129, 205, 493	81	189, 237, 243, 781, 1357, 1537	117	297, 333, 565, 1177, 1717, 2581, 3337
10		46	66, 70	82	130	118	174, 190
11	35, 121	47	215, 287, 407, 527, 551, 2209	83	395, 803, 923, 1139, 1403, 1643, 1739, 1763, 6889	119	539, 791, 1199, 1391, 1751, 1919, 2231, 2759, 3071, 3239, 3431, 3551, 3599
12	18, 20, 22	48	72, 80, 88, 92, 94	84	164, 166	120	168, 200, 232, 236
13	33, 169	49	141, 301, 343, 481, 589	85	165, 249, 325, 553, 949, 1273	121	1331, 1417, 1957, 3397
14	26	50		86		122
15	39, 55	51	235, 451, 667	87	415, 1207, 1711, 1927	123	1243, 1819, 2323, 3403, 3763
16	24, 28, 32	52		88	120, 172	124	244
17	65, 77, 289	53	329, 473, 533, 629, 713, 2809	89	581, 869, 1241, 1349, 1541, 1769, 1829, 1961, 2021, 7921	125	625, 1469, 1853, 2033, 2369, 2813, 3293, 3569, 3713, 3869, 3953
18	34	54	78, 106	90	126, 178	126	186
19	51, 91, 361	55	159, 175, 559, 703	91	267, 1027, 1387, 1891	127	255, 2071, 3007, 4087, 16129
20	38	56	98, 104	92	132, 140	128	192, 224, 248, 254, 256
21	45, 57, 85	57	105, 153, 265, 517, 697	93	261, 445, 913, 1633, 2173	129	273, 369, 381, 1921, 2461, 2929, 3649, 3901, 4189
22	30	58		94	138, 154	130
23	95, 119, 143, 529	59	371, 611, 731, 779, 851, 899, 3481	95	623, 1079, 1343, 1679, 1943, 2183, 2279	131	635, 2147, 2507, 2987, 3131, 3827, 4187, 4307, 4331, 17161
24	36, 40, 44, 46	60	84, 100, 116, 118	96	144, 160, 176, 184, 188	132	180, 242, 262
25	69, 125, 133	61	177, 817, 3721	97	1501, 2077, 2257, 9409	133	393, 637, 889, 3193, 3589, 4453
26		62	122	98	194	134
27	63, 81, 115, 187	63	135, 147, 171, 183, 295, 583, 799, 943	99	195, 279, 291, 979, 1411, 2059, 2419, 2491	135	351, 387, 575, 655, 2599, 3103, 4183, 4399
28	52	64	96, 112, 124, 128	100		136	268
29	161, 209, 221, 841	65	305, 413, 689, 893, 989, 1073	101	485, 1157, 1577, 1817, 2117, 2201, 2501, 2537, 10201	137	917, 1397, 3161, 3317, 3737, 3977, 4661, 4757, 18769
30	42, 50, 58	66	90	102	202	138	198, 274
31	87, 247, 961	67	427, 1147, 4489	103	303, 679, 2263, 2479, 2623, 10609	139	411, 1651, 3379, 3811, 4171, 4819, 4891, 19321
32	48, 56, 62, 64	68	134	104	206	140	204, 220, 278
33	93, 145, 253	69	201, 649, 901, 1081, 1189	105	225, 309, 425, 505, 1513, 1909, 2773	141	285, 417, 685, 1441, 3277, 4141, 4717, 4897
34		70	102, 110	106	170	142	230, 238
35	75, 155, 203, 299, 323	71	335, 671, 767, 1007, 1247, 1271, 5041	107	515, 707, 1067, 1691, 2291, 2627, 2747, 2867, 11449	143	363, 695, 959, 1703, 2159, 3503, 3959, 4223, 4343, 4559, 5063, 5183
36	54, 68	72	108, 136, 142	108	156, 162, 212, 214	144	216, 272, 284

References

Browkin, J.; Schinzel, A. (1995). "On integers not of the form n-φ(n)". Colloq. Math. 68 (1): 55–58. Zbl 0820.11003.
Flammenkamp, A.; Luca, F. (2000). "Infinite families of noncototients". Colloq. Math. 86 (1): 37–41. Zbl 0965.11003.
Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 138–142. ISBN 978-0-387-20860-2. Zbl 1058.11001.

External links

Noncototient definition from MathWorld

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