cp's OEIS Frontend

It is interesting that most of the numbers have the last digit 1. For example 530881, 3581761, 7207201, etc.

Granville & Pomerance conjecture that there are ~ c x^(1/3)/(log x)^3 terms of this sequence up to x. Heath-Brown proves that, for any e > 0, there are O(x^(7/20 + e)) terms of this sequence up to x. - Charles R Greathouse IV, Nov 19 2012

All 3-term Carmichael numbers can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions. Under Dickson's conjecture, "almost all" 3-term Carmichael numbers are primary Carmichael numbers (A324316) in the sense that C'3(x)/C_3(x) -> 1 as x -> infinity, where C_3(x) counts the 3-term Carmichael numbers and C'_3(x) counts the 3-term primary Carmichael numbers up to x. A primary Carmichael number m has the unique property (*) that for each prime divisor p of m, the sum of the base-p digits of m equals p. As a nontrivial result, all 3-term Carmichael numbers m have at least the property that (*) holds for the greatest prime divisor p of m, see Kellner 2019. - _Bernd C. Kellner, Aug 03 2022

Also called Chernick's Carmichael numbers. The polynomial (6*k+1)*(12*k+1)*(18*k+1) is the simplest Chernick polynomial. [Named after the American physicist and mathematician Jack Chernick (1911-1971). - Amiram Eldar, Jun 15 2021]

The first term, 1729, is the Hardy-Ramanujan number and the smallest primary Carmichael number (A324316).

Dickson's conjecture implies that this sequence is infinite, as pointed out by Chernick.

All terms of this sequence are primary Carmichael numbers (A324316) having the following remarkable property. Let m be a term of A033502. For each prime divisor p of m, the sum of the base-p digits of m equals p. This property also holds for "almost all" 3-term Carmichael numbers (A087788), since they can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions, see Kellner 2019. - Bernd C. Kellner, Aug 03 2022

The function s_g(m) gives the sum of the base-g digits of m.

The sequence is infinite, since it contains A324460.

The sequence also contains the 3-Carmichael numbers A087788 and the primary Carmichael numbers A324316.

A term m must have at least 2 prime factors, and the divisor g satisfies the inequalities 1 < g < m^(1/(ord_g(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.

Note that the sequence contains the 3-Carmichael numbers, but not all Carmichael numbers. This is a nontrivial fact.

The subsequence A324460 mainly gives examples in which g is composite.

See Kellner 2019.

It appears that g is usually prime: compare with A324857 (g prime) and the sparser sequence A324858 (g composite). However, g is usually composite for higher values of m. - Jonathan Sondow, Mar 17 2019

The function s_g(m) gives the sum of the base-g digits of m.

The sequence is infinite, since it contains A324460 and the Carmichael numbers A002997.

A term m must have at least 2 prime factors, and the divisor g satisfies the inequalities 1 < g < m^(1/(ord_g(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.

See Kellner 2019.

The function s_p(m) gives the sum of the base-p digits of m.

The sequence is infinite, since it contains A324315, and thus the Carmichael numbers A002997.

Being a subsequence of A324459, a term m has the following properties:

m must have at least 2 prime factors. If m = p1^e1 * p2^e2 with two primes p1 and p2, then e1 + e2 >= 3.

Each prime factor p of m satisfies the inequalities p < m^(1/(ord_p(m)+1)) <= sqrt(m), where ord_p(m) gives the maximum exponent e such that p^e divides m.

In the terminology of A324459, the prime factorization of m equals an s-decomposition of m.

See Kellner 2019.

a(n) is a Carmichael number A002997 iff a(n) is squarefree and s_p(a(n)) == 1 (mod p-1) for every prime factor p of a(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 16 2019

The function s_p(m) gives the sum of the base-p digits of m.

The sequence contains the primary Carmichael numbers A324316.

Being a subsequence of A324460, a term m has the following properties:

m must have at least 2 prime factors. If m = p1^e1 * p2^e2 with two primes p1 and p2, then e1 + e2 >= 3.

Each prime factor p of m satisfies the inequalities p < m^(1/(ord_p(m)+1)) <= sqrt(m), where ord_p(m) gives the maximum exponent e such that p^e divides m.

In the terminology of A324460, the prime factorization of m equals a strict s-decomposition of m.

See Kellner 2019.

a(n) is squarefree iff a(n) is a primary Carmichael number A324316. - Jonathan Sondow, Mar 16 2019

The sequence is infinite, since it contains A324460 and the Carmichael numbers A002997.

A number m > 1 has an s-decomposition if there exists a decomposition in n proper factors g_k with exponents e_k >= 1 (the factors g_k being strictly increasing but not necessarily coprime) such that

m = g_1^e_1 * ... * g_n^e_n, where s_{g_k}(m) >= g_k for all k,

and s_g(m) gives the sum of the base-g digits of m.

A term m has the following properties:

m must have at least 2 factors g_k. If m = g_1^e_1 * g_2^e_2 with exactly two factors, then e_1 + e_2 >= 3.

Each factor g_k of m satisfies the inequalities 1 < g_k < m^(1/(ord_{g_k}(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.

See Kellner 2019.