A095347 -id:A095347 - cp's OEIS frontend

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

Jason Earls, Aug 04 2003; revised Jun 03 2004

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

			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.

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.]

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)

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

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

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

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 *)

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

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.

A094501 Smallest number that requires n iterations of the sum of digits of the divisors (A034690) to reach 15.

Original entry on oeis.org

15, 8, 7, 4, 3, 2, 19, 12, 6, 5, 13, 9, 10, 16, 30, 18, 34, 36, 66, 162, 924, 71820, 127005777360

Offset: 0

Jason Earls, Jun 05 2004

			a(0)=15 trivially because 15 is reached in no steps (number of steps is 0);
a(1)=8 because divisors of 8 are 1,2,4,8 with sum of digits = 15 hence 15 is reached in 1 steps (number of steps is 1);
a(2)=7 because divisors of 7 are 1,7 with sum of digits =8 and we need another one step to reach 15 (number of steps is 2);
a(3)=4 because divisors of 4 are 1,2,4 with sum of digits =7 and we need another two steps to reach 15 (number of steps is 3);
a(20)=924 because starting with 924 we have the trajectory 924, 168, 102, 36, 46, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15 reaching 15 in 20 steps.
a(21)=71820 because starting with 71820 we have the trajectory 71820, 1104, 168, 102, 36, 46, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15 reaching 15 in 21 steps. - _Sean A. Irvine_, Oct 04 2009

Eric Angelini et al., List the dividers [sic], sum the digits, lost messages reconstructed by _N. J. A. Sloane_, Dec 21 2024

Cf. A086793, A095347, A119397.

See A260060 for another variant.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a094501 = (+ 2) . fromJust . (`elemIndex` a086793_list)
-- Reinhard Zumkeller, Nov 08 2015

f[n_] := Block[{i = 0}, NestWhile[(i++; Plus @@ Flatten@ IntegerDigits@ Divisors@#) &, n, # != 15 &]; i]; t = Table[0, {100}]; Do[ a = f[n]; If[ t[[a]] < 101 && t[[a]] == 0, t[[a]] = n], {n, 2, 10^8}]; t (* Robert G. Wilson v, May 16 2006 *)

A094501(n)=for(k=2, 9e9, A086793(k)==n&&return(k)) \\ M. F. Hasler, Nov 08 2015

Examples provided by Zak Seidov, May 16 2006
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 10 2007
a(22) found by exhaustive search by Sean A. Irvine, Oct 04 2009
a(22) corrected by Donovan Johnson and Sean A. Irvine

A119396 Numbers n such that A086793(n)=20.

Original entry on oeis.org

924, 1104, 1134, 1540, 1650, 1760, 1820, 1908, 1992, 2016, 2288, 2556, 2632, 2744, 2860, 2940, 2970, 3000, 3192, 3204, 3220, 3248, 3400, 3630, 3738, 3784, 3840, 3852, 3880, 3968, 3990, 4134, 4260, 4410, 4464, 4674, 4736, 4860, 4875, 4930, 4992, 5016

Offset: 1

Zak Seidov, May 17 2006

Some trajectories are: 924,168,102,36,46,18,30,27,22,9,13,5,6,12,19,11,3,4,7,8,15 1104,168,102,... 1540,162,66,36,... 1650,162,66,36,... 2016,297,66,36,... 2940,297,66,36,... 3192,312,102,36,... All trajectories eventually join one of previous trajectories.

			924 is a term because it reaches 15 in 20 steps with this trajectory 924,168,102,36,46,18,30,27,22,9,13,5,6,12,19,11,3,4,7,8,15.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034690, A086793, A094501, A095347, A114527, A118358, A119397, A119398, A260059.

f:= proc(n) option remember; local t;
  if kernelopts(level) > 460 then return FAIL fi;
  t:= add(convert(convert(d,base,10),`+`),d=numtheory:-divisors(n));
  1+procname(t)
end proc:
f(15):= 0:
f(1):= FAIL:
Res:= NULL: count:= 0:
for n from 1 while count < 100 do
  if f(n) = 20 then
    count:= count+1;
    Res:= Res, n;
   fi
od:
Res; # Robert Israel, Apr 03 2018

Edited by Robert Israel, Apr 03 2018

A118358 Records in A086793.

Original entry on oeis.org

5, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22

Offset: 1

N. J. A. Sloane, May 15 2006

Cf. A086793. A095347 is where these records happen.

a(13) from Donovan Johnson, Dec 23 2010

A260060 Least number such that exactly n iterations of A034690 are required to reach one of the fixed points, 1 or 15.

Original entry on oeis.org

1, 8, 7, 4, 3, 2, 19, 12, 6, 5, 13, 9, 10, 16, 30, 18, 34, 36, 66, 162, 924, 71820

Offset: 0

M. F. Hasler, Nov 08 2015

Apart from the initial term a(1), the same as A094501.

			The orbits are:
  {1},
  {8, 15},
  {7, 8, 15},
  {4, 7, 8, 15},
  {3, 4, 7, 8, 15},
  {2, 3, 4, 7, 8, 15},
  {19, 11, 3, 4, 7, 8, 15},
  {12, 19, 11, 3, 4, 7, 8, 15},
  {6, 12, 19, 11, 3, 4, 7, 8, 15},
  {5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {10, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {16, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {34, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {36, 46, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {66, 36, 46, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {162, 66, 36, 46, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {924, 168, 102, 36, 46, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15},
  {71820, 1104, 168, 102, 36, 46, 18, 30, 27, 22, 9, 13, 5, 6, 12, 19, 11, 3, 4, 7, 8, 15}

E. 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

Cf. A034690, A086793, A094501, A095347.

a(n)=for(k=1,9e9, A086793(k)==n&&return(k))

A373094 a(n) is the least number k such that A373092(k) = n.

Original entry on oeis.org

1, 4, 7, 12, 24, 120, 1260, 1829520

Offset: 0

Amiram Eldar, May 23 2024

a(n) is the least number k such that the number of iterations of the map x -> A093653(x) required to reach from k to a fixed point is n.

a(8) > 4*10^10.

			The iterations for the n = 0..7 are:
  n     a(n)  iterations
  -  -------  --------------------------------------------------
  0        1   1
  1        4   4 -> 3
  2        7   7 -> 4 -> 3
  3       12   12 -> 9 -> 5 ->3
  4       24   24 -> 12 -> 9 -> 5 -> 3
  5      120   120 -> 36 -> 15 -> 9 -> 5 -> 3
  6     1260   1260 -> 120 -> 36 -> 15 -> 9 -> 5 -> 3
  7  1829520   1829520 -> 1260 -> 120 -> 36 -> 15 -> 9 -> 5 -> 3

Cf. A093653, A095347 (decimal analog), A373092.

d[n_] := d[n] = DivisorSum[n, Plus @@ IntegerDigits[#, 2] &];
f[n_] := -2 + Length@ FixedPointList[d, n];
seq[len_] := Module[{s = Table[0, {len}], c = 0, i, n = 1}, While[c < len, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[7]

f(n) = {my(c = 0); while(6 % n, n = sumdiv(n, d, hammingweight(d)); c++); c;}
lista(len) = {my(s = vector(len), c = 0, i, n = 1); while(c < len, i = f(n) + 1; if(i <= len && s[i] == 0, c++; s[i] = n); n++); s;}

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Extensions

A094501 Smallest number that requires n iterations of the sum of digits of the divisors (A034690) to reach 15.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A119396 Numbers n such that A086793(n)=20.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A118358 Records in A086793.

Views

Author

Keywords

Crossrefs

Extensions

A260060 Least number such that exactly n iterations of A034690 are required to reach one of the fixed points, 1 or 15.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A373094 a(n) is the least number k such that A373092(k) = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI