OFFSET
1,1
COMMENTS
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
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..806 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
6 is a member, since 2 divides 6 and s_2(6) = 2.
MATHEMATICA
s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
f[n_] := AnyTrue[Divisors[n], s[n, #] == # &];
Select[Range[5000], f[#] &]
PROG
(PARI) isok(n) = {fordiv(n, d, if ((d>1) && (sumdigits(n, d) == d), return (1)); ); } \\ Michel Marcus, Mar 19 2019
CROSSREFS
KEYWORD
nonn,base,changed
AUTHOR
Bernd C. Kellner, Feb 28 2019
STATUS
approved