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 A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g. 8
 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 (list; graph; refs; listen; history; text; internal format)
 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, 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, 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 Subsequences are A033502, A087788, A324316, A324458, A324460. Subsequence of A324455. Cf. A324315, A324457, A324459. Union of A324857 and A324858. Sequence in context: A210578 A274426 A345995 * A324857 A279550 A362754 Adjacent sequences: A324453 A324454 A324455 * A324457 A324458 A324459 KEYWORD nonn,base AUTHOR Bernd C. Kellner, Feb 28 2019 STATUS approved

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Last modified December 10 18:13 EST 2023. Contains 367717 sequences. (Running on oeis4.)