A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

6, 10, 12, 15, 18, 20, 21, 24, 28, 33, 34, 36, 39, 40, 45, 48, 52, 57, 63, 65, 66, 68, 72, 76, 80, 85, 87, 88, 91, 93, 96, 99, 100, 105, 111, 112, 117, 120, 126, 130, 132, 133, 135, 136, 144, 145, 148, 153, 156, 160, 165, 171, 175, 176, 185, 186, 189, 190

Offset: 1

Author

Bernd C. Kellner, Feb 28 2019

Keywords

• nonn
• base

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

Examples

			6 is a member, since 2 divides 6 and s_2(6) = 2.
		

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.

Crossrefs

Subsequences are A033502, A087788, A324316, A324458, A324460.

Subsequence of A324455.

Cf. A324315, A324457, A324459.

Union of A324857 and A324858.

Programs

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[#] &]

PARI

isok(n) = {fordiv(n, d, if ((d>1) && (sumdigits(n, d) == d), return (1)););} \\ Michel Marcus, Mar 19 2019