cp's OEIS Frontend

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

%I A064807 #60 Dec 08 2024 20:41:44
%S A064807 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,
%T A064807 46,48,50,54,55,56,57,60,63,64,66,70,72,73,74,75,76,78,80,81,82,84,90,
%U A064807 91,92,93,95,96,99,100,102,108,109,110,111,112,114,117,118
%N A064807 Numbers which are divisible by their digital root (A010888).
%C A064807 All numbers 9m, m > 0, belong to this sequence.
%C A064807 All numbers 6m, m > 0, belong to this sequence. - _Christian Schulz_, Oct 30 2013
%C A064807 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
%C A064807 Conjecture: All k-multiply perfect numbers belong to this sequence. - _Ivan N. Ianakiev_, May 10 2016
%C A064807 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
%C A064807 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
%H A064807 Ray Chandler, <a href="/A064807/b064807.txt">Table of n, a(n) for n = 1..4196</a> (first 1000 terms from Harry J. Smith)
%H A064807 <a href="/index/Rec#order_1322">Index entries for linear recurrences with constant coefficients</a>, order 1322.
%F A064807 a(n) = a(n-1321) + 2520. - _Charles R Greathouse IV_, Dec 26 2013
%F A064807 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
%F A064807 a(n) = a(n-1) + a(n-1321) - a(n-1322). - _Charles R Greathouse IV_, Apr 20 2023
%e A064807 48: 4 + 8 = 12 -> 1 + 2 = 3. 48 = 3 * 16 therefore 48 = a(28).
%p A064807 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
%t A064807 Select[Range[125], Divisible[#, Mod[# - 1, 9] + 1] &] (* _Alonso del Arte_, Nov 01 2013 *)
%o A064807 (PARI) is(n)=n%((n-1)%9+1)==0 \\ _Charles R Greathouse IV_, Dec 26 2013
%o A064807 (Haskell)
%o A064807 a064807 n = a064807_list !! (n-1)
%o A064807 a064807_list = filter (\x -> x `mod` a010888 x == 0) [1..]
%o A064807 -- _Reinhard Zumkeller_, Jan 03 2014
%Y A064807 Cf. A010888, A074947, A074949.
%K A064807 nonn,base,easy
%O A064807 1,2
%A A064807 _Reinhard Zumkeller_, Oct 21 2001
%E A064807 Offset changed from 0 to 1 by _Harry J. Smith_, Sep 26 2009