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

Author

Jason Earls, Aug 04 2003; revised Jun 03 2004

Keywords

• base
• easy
• nonn

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

Examples

			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.
		

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 [Broken link]
• Eric Angelini et al., List the dividers [sic], sum the digits, lost messages reconstructed by _N. J. A. Sloane_, Dec 21 2024
• Michael W. Ecker, Divisive Number 15. (Web archive, as of May 2008)

Crossrefs

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

Programs

Haskell

a086793 = f 0 where
   f y x = if x == 15 then y else f (y + 1) (a034690 x)
-- Reinhard Zumkeller, May 20 2015

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 *)

PARI

A086793(n)=n>1&&for(k=0, oo, n==15&&return(k); n=A034690(n)) \\ M. F. Hasler, Nov 08 2015

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.