

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
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OFFSET

1,1


COMMENTS

The function s_g(m) gives the sum of the baseg digits of m.
The sequence is infinite, since it contains A324460.
The sequence also contains the 3Carmichael 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 3Carmichael 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



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


AUTHOR



STATUS

approved



