A086793 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.

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, 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, 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

Offset

1,2

Comments

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]

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

References

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.]

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (corrected by Georg Fischer, Jan 20 2019)

Eric Angelini et al., List the dividers, sum the digits, SeqFan list, Nov. 2015

Michael W. Ecker, Divisive Number 15. (Web archive, as of May 2008)

Example

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.

Maple

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;
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

Mathematica

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 *)

Prog

(Haskell)
a086793 = f 0 where
   f y x = if x == 15 then y else f (y + 1) (a034690 x)
-- Reinhard Zumkeller, May 20 2015
(PARI) A086793(n)=n>1&&for(k=0, 9e9, n==15&&return(k); n=A034690(n)) \\ M. F. Hasler, Nov 08 2015

Crossrefs

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

Sequence in context: A202412 A194556 A154198 * A070515 A096733 A125900

Adjacent sequences: A086790 A086791 A086792 * A086794 A086795 A086796

Keyword

base,easy,nonn

Author

Jason Earls, Aug 04 2003; revised Jun 03 2004

Extensions

Corrected by N. J. A. Sloane, May 17 2006 (a(15) changed to 0)

Corrected by David Applegate, Jan 23 2007 (reference book title corrected)

Extended to a(1)=0 by M. F. Hasler, Nov 08 2015

Status

approved