A086793 - OEIS

%I #53 Dec 23 2024 14:53:42

%S 0,5,4,3,9,8,2,1,11,12,5,7,10,1,0,13,12,15,6,1,2,12,9,9,11,1,13,9,8,

%T 14,10,14,8,16,3,17,6,10,2,14,9,9,2,3,9,16,8,3,3,3,16,2,12,4,16,4,2,

%U 14,1,10,2,1,15,7,3,18,2,18,10,18,12,11,6,10,17,10,10,17,13,10,11,16,8,2,14,10,15

%N Number of iterations of the map A034690 (x -> sum of digits of all divisors of x) required to reach one of the fixed points, 15 or 1.

%C Ecker states that every number (larger than 1) eventually reaches 15. "Take any natural number larger than 1 and write down its divisors, including 1 and the number itself. Now take the sum of the digits of these divisors. Iterate until a number repeats. The black-hole number this time is 15." [Ecker]

%C The only other fixed point of A034690, namely 1, cannot be reached by any other starting value than 1 itself. - _M. F. Hasler_, Nov 08 2015

%D Michael W. Ecker, Number play, calculators and card tricks ..., pp. 41-51 of The Mathemagician and the Pied Puzzler, Peters, Boston. [Suggested by a problem in this article.]

%H Reinhard Zumkeller, <a href="/A086793/b086793.txt">Table of n, a(n) for n = 1..10000</a> (corrected by _Georg Fischer_, Jan 20 2019)

%H Eric Angelini et al., <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-November/015585.html">List the dividers, sum the digits</a>, SeqFan list, Nov. 2015 [Broken link]

%H Eric Angelini et al., <a href="/A086793/a086793.txt">List the dividers [sic], sum the digits</a>, lost messages reconstructed by _N. J. A. Sloane_, Dec 21 2024

%H Michael W. Ecker, <a href="http://web.archive.org/web/20080515203911/http://members.aol.com/DrMWEcker/mathhole.html">Divisive Number 15</a>. (Web archive, as of May 2008)

%e 35 requires 3 iterations to reach 15 because 35 -> 1+5+7+3+5 = 21 -> 1+3+7+2+1 = 14 -> 1+2+7+1+4 = 15.

%p with(numtheory); read transforms; f:=proc(n) local t1,t2,i; t1:=divisors(n); t2:=0; for i from 1 to nops(t1) do t2:=t2+digsum(t1[i]); od: t2; end;

%p g:=proc(n) global f; local t2,i; t2:=n; for i from 1 to 100 do if t2 = 15 then return(i-1); fi; t2:=f(t2); od; end; # _N. J. A. Sloane_

%t f[n_] := (i++; Plus @@ Flatten@IntegerDigits@Divisors@n); Table[i = 0; NestWhile[f, n, # != 15 &]; i, {n, 2, 87}] (* _Robert G. Wilson v_, May 16 2006 *)

%o (Haskell)

%o a086793 = f 0 where

%o    f y x = if x == 15 then y else f (y + 1) (a034690 x)

%o -- _Reinhard Zumkeller_, May 20 2015

%o (PARI) A086793(n)=n>1&&for(k=0, 9e9, n==15&&return(k); n=A034690(n)) \\ _M. F. Hasler_, Nov 08 2015

%Y Cf. A034690, A114527. For records see A095347, A118358.

%K base,easy,nonn

%O 1,2

%A _Jason Earls_, Aug 04 2003; revised Jun 03 2004

%E Corrected by _N. J. A. Sloane_, May 17 2006 (a(15) changed to 0)

%E Corrected by _David Applegate_, Jan 23 2007 (reference book title corrected)

%E Extended to a(1)=0 by _M. F. Hasler_, Nov 08 2015.