Table 1. Weak Carmichael numbers up to 25000 (the sequence

Table 1. Weak Carmichael numbers up to 25000 (the sequence A225498 in The-On-Line-Encyclopedia of Integer Sequences (OEIS), available at https//oeis.org/A225498) 9 = 32 25 = 5 2 27 = 3 3 45 = 32 · 5 49 = 7 2 81 = 3 4 121 = 11 2 125 = 5 3 2 2 2 5 169 = 13 225 = 3 · 5 243 = 3 289 = 17 2 2 3 2 325 = 5 · 13 343 = 7 361 = 19 405 = 34 · 5 2 4 529 = 23 561 = 3 · 11 · 17 625 = 5 637 = 72 · 13 729 = 3 6 841 = 29 2 891 = 34 · 11 961 = 31 2 2 3 2 2 1105 = 5 · 13 · 17 1125 = 3 · 5 1225 = 5 · 7 1331 = 11 3 2 4 2 1369 = 37 1377 = 3 · 17 1681 = 41 1729 = 7 · 13 · 19 1849 = 43 2 2025 = 34 · 52 2187 = 3 7 2197 = 13 3 2209 = 47 2 2401 = 7 4 2465 = 5 · 17 · 29 2809 = 53 2 5 4 2821 = 7 · 13 · 31 3125 = 5 3321 = 3 · 41 3481 = 59 2 6 2 2 3645 = 3 · 5 3721 = 61 3751 = 11 · 31 3825 = 32 · 52 · 17 2 2 2 3 4225 = 5 · 13 4489 = 67 4913 = 17 4961 = 112 · 41 5041 = 71 2 5329 = 73 2 5589 = 35 · 23 5625 = 32 · 54 2 3 2 2 6241 = 79 6517 = 7 · 19 6525 = 3 · 5 · 29 6561 = 3 8 3 2 6601 = 7 · 23 · 41 6859 = 19 6889 = 83 7381 = 112 · 61 2 4 2 2 7921 = 89 8125 = 5 · 13 8281 = 7 · 13 8625 = 3 · 53 · 23 8911 = 7 · 19 · 67 9409 = 97 2 9801 = 34 · 112 10125 = 34 · 53 2 2 10201 = 101 10585 = 5 · 29 · 73 10609 = 103 10625 = 54 · 17 2 2 2 11449 = 107 11881 = 109 12025 = 5 · 13 · 37 12167 = 23 3 2 2 2 12769 = 113 13357 = 19 · 37 13833 = 3 · 29 · 53 14161 = 72 · 172 14641 = 11 4 15625 = 5 6 15841 = 7 · 31 · 73 15925 = 52 · 72 · 13 2 5 16129 = 127 16807 = 7 17161 = 131 2 18225 = 36 · 52 2 2 9 18769 = 137 19321 = 139 19683 = 3 21141 = 36 · 29 2 2 2 2 22201 = 149 22801 = 151 23409 = 3 · 5 · 23 23805 = 32 · 5 · 232 24389 = 29 3 24649 = 157 2

Remark. Notice that the notion of weak Carmichael numbers was introduced in the article by Romeo Meˇstrovi´c, Generalizations of Carmichael numbers I, preprint arXiv:1305.1867v1 [math.NT], 4 May 2013, 46 pages. Accordingly, a composite positive integer n is said to be a weak Carmichael number if X k n−1 ≡ ϕ(n) (mod n), gcd(k,n)=1 1≤k≤n−1

where gcd(k, n) denotes the greatest common divisor of k and n, and ϕ(n) is the Euler totient function, defined as the number of positive integers less than n which are relatively prime to n. The above Table 1 was given as Table 1 on page 15 in the previously mentioned paper. Table 1 shows that there are 102 weak Carmichael numbers less than 25000, and between them there are 9 Carmichael numbers, 57 odd prime powers and 36 other composite numbers. In bold are assigned Carmichael numbers. Recall that in 2006 R.G.E. Pinch (The Carmichael numbers up to 1

2

1018 ; preprint arXiv:math/0604376v1 [math.NT], 2006) reported that there are 1401644 Carmichael numbers up to 1018 .