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