A064807 Numbers which are divisible by their digital root (A010888).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 19, 20, 21, 24, 27, 28, 30, 36, 37, 38, 39, 40, 42, 45, 46, 48, 50, 54, 55, 56, 57, 60, 63, 64, 66, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 90, 91, 92, 93, 95, 96, 99, 100, 102, 108, 109, 110, 111, 112, 114, 117, 118

Offset: 1

Author

Reinhard Zumkeller, Oct 21 2001

Keywords

• nonn
• base
• easy

Comments

All numbers 9m, m > 0, belong to this sequence.

All numbers 6m, m > 0, belong to this sequence. - Christian Schulz, Oct 30 2013

All numbers 280m, m > 0, belong to this sequence. Only 6, 9, 280, and their multiples have this property. - Charles R Greathouse IV, Dec 26 2013

Conjecture: All k-multiply perfect numbers belong to this sequence. - Ivan N. Ianakiev, May 10 2016

The asymptotic density of this sequence is 1321/2520 = 0.524206... (see A074947 and A074949 for the values in other base representations). - Amiram Eldar, Nov 24 2022

The even perfect numbers are a subsequence. It is an open question whether the odd perfect numbers are a subsequence; this would involve ruling out 148 residue classes mod 2520 as OPNs. - Charles R Greathouse IV, Jan 03 2023

Examples

			48: 4 + 8 = 12 -> 1 + 2 = 3. 48 = 3 * 16 therefore 48 = a(28).
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..4196 (first 1000 terms from Harry J. Smith)
• Index entries for linear recurrences with constant coefficients, order 1322.

Crossrefs

Cf. A010888, A074947, A074949.

Programs

Haskell

a064807 n = a064807_list !! (n-1)
a064807_list = filter (\x -> x `mod` a010888 x == 0) [1..]
-- Reinhard Zumkeller, Jan 03 2014

Maple

A064807 := proc(n) option remember: local k: if(n=1)then return 1:fi: for k from procname(n-1)+1 do if(k mod (((k-1) mod 9) + 1) = 0)then return k: fi: od: end: seq(A064807(n),n=1..100); # Nathaniel Johnston, May 05 2011

Mathematica

Select[Range[125], Divisible[#, Mod[# - 1, 9] + 1] &] (* Alonso del Arte, Nov 01 2013 *)

PARI

is(n)=n%((n-1)%9+1)==0 \\ Charles R Greathouse IV, Dec 26 2013

Formula

a(n) = a(n-1321) + 2520. - Charles R Greathouse IV, Dec 26 2013

2520n/1321 - 10 < a(n) <= 2520n/1321. (In fact, if you exclude n = 10 mod 1321, you can replace 10 with 9.) - Charles R Greathouse IV, Jan 03 2023

a(n) = a(n-1) + a(n-1321) - a(n-1322). - Charles R Greathouse IV, Apr 20 2023

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Sep 26 2009