OFFSET
1,1
COMMENTS
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
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..117 from Bernd C. Kellner)
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
EXAMPLE
The number 45 has the prime factors 3 and 5. Since s_3(45) = 3 and s_5(45) = 5, 45 is a member.
MATHEMATICA
s[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
f[n_] := AllTrue[Transpose[FactorInteger[n]][[1]], s[n, #] == # &];
Select[Range[10^7], f[#] &]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Bernd C. Kellner, Feb 28 2019
STATUS
approved