A114527 -id:A114527 - 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.

A106756 Primes with digit sum = 14.

Original entry on oeis.org

59, 149, 167, 239, 257, 293, 347, 383, 419, 491, 509, 563, 617, 653, 743, 761, 941, 1049, 1193, 1229, 1283, 1319, 1373, 1409, 1427, 1481, 1553, 1571, 1607, 1733, 1823, 1913, 1931, 2039, 2129, 2237, 2273, 2309, 2381, 2417, 2543, 2633, 2741, 2903, 3083

Offset: 1

Zak Seidov, May 16 2005

Or prime numbers in A114527. - Zak Seidov, May 21 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040 (primes), A007953 (sum of digits), A235225 (digit sum = 14).
Cf. A062339 (same for digit sum s = 4), A106755 (s = 13), A106757 (s = 16), and others listed in A244918 (s = 68).
Cf. A034690, A086793, A114527.

[p: p in PrimesUpTo(4000) | &+Intseq(p) eq 14]; // Vincenzo Librandi, Jul 08 2014

Select[Prime[Range[10000]], Total[IntegerDigits[#]]==14 &] (* Vincenzo Librandi, Jul 08 2014 *)

select( {is_A106756(n)= sumdigits(n)==14 && isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A235225 (digit sum = 14); also equals { p in A000040 | A007953(p) = 14 }. - M. F. Hasler, Mar 09 2022

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

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

A106756 Primes with digit sum = 14.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions