A333457 a(n) is the smallest number with exactly n divisors that are Moran numbers, or -1 if no such number exists.

18, 42, 84, 126, 252, 756, 1998, 1596, 2394, 4662, 4788, 9324, 18648, 23940, 46620, 93240, 139860, 177156, 559440, 354312, 708624, 1062936, 885780, 4606056, 1771560, 3543120, 5314680, 10629360, 38974320, 23030280, 46060560, 69090840, 138181680, 506666160

Offset

1,1

Comments

m is a Moran number if (m / sum of digits of m) is prime (A001101).

Conjecture: For every n there is at least one number k with n divisors Moran numbers.

Conjecture: The terms are divisible by 6.

a(1) = 18, a(2) = 42 and a(3) = 84 are Moran numbers. Not all terms in the sequence are Moran numbers. For example: a(4) = 126 has digsum(126) = 9 and 126 / 9 = 14. Also, the terms a(5) - a(34) are not Moran numbers.

Links

Table of n, a(n) for n=1..34.

Example

Of the divisors of 18 (1, 2, 3, 6, 9, 18), only 18 is a Moran number: 18 / digsum (18) = 2.
Of the divisors of 84 (1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84), only 21, 42 and 84 are Moran numbers: 21 / digsum (21) = 7, 42 / digsum (42) = 7 and 84 / digsum (84) = 7.

Mathematica

numDiv[n_] := DivisorSum[n, 1 &, PrimeQ[#/Plus @@ IntegerDigits[#]] &]; a[n_] := Module[{k = 1}, While[numDiv[k] != n, k++]; k]; Array[a, 20] (* Amiram Eldar, May 11 2020 *)

Prog

(Magma) a:=[]; for n in [1..20] do m:=1; while #[d:d in Divisors(m)|d mod &+Intseq(d) eq 0 and IsPrime(d div &+Intseq(d))] ne n do m:=m+1; end while; Append(~a, m); end for; a;

Crossrefs

Cf. A001101, A007953 (sum of digits), A333456.

Sequence in context: A175152 A105550 A144737 * A304152 A252747 A067750

Adjacent sequences: A333454 A333455 A333456 * A333458 A333459 A333460

Keyword

nonn,base

Author

Marius A. Burtea, May 03 2020

Status

approved