A086793 -id:A086793 - cp's OEIS frontend

A095347 n sets a new record for number of iterations of A034690 (sum of digits of the divisors of n) needed to reach 15 (see A086793).

Original entry on oeis.org

2, 5, 9, 10, 16, 18, 34, 36, 66, 162, 924, 71820, 127005777360

Offset: 1

Jason Earls, Jun 03 2004

323203999999676796 takes 22 iterations to reach 15, but it probably is not the next term.

One could prefix a(0)=1 and change the definition to "... reach a fixed point, 1 or 15." - M. F. Hasler, Nov 08 2015

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.

m=0;for(n=1,9e9, A086793(n)>m||next;m=A086793(n);print1(n","))

Offset corrected and a(13) from Donovan Johnson, Oct 28 2010

A114527 Numbers k such that A086793(k) is 1.

Original entry on oeis.org

8, 14, 20, 26, 59, 62, 122, 123, 143, 149, 167, 206, 239, 257, 293, 302, 341, 347, 383, 419, 422, 491, 509, 563, 617, 653, 743, 761, 941, 1049, 1133, 1193, 1202, 1203, 1229, 1283, 1313, 1319, 1331, 1373, 1409, 1427, 1481, 1553, 1571, 1607

Offset: 1

Zak Seidov, May 16 2006

Prime numbers in the sequence are also primes with digit sum = 14 (A106756). - Zak Seidov, May 21 2006

Cf. A034690, A086793, A106756.

ss={8,14};Do[If[15==Total@Flatten[IntegerDigits/@Divisors[n]],AppendTo[ss,n]],{n,20,2000}];ss (* Zak Seidov, May 21 2006 *)

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

A119398 Odd numbers taking exactly 21 steps to reach 15 in A086793.

Original entry on oeis.org

628425, 824175, 1340325, 1422135, 1495725, 1729665, 1845585, 1853775, 1916145, 2001825, 2015685, 2040675, 2045505, 2091375, 2165625, 2220435, 2226609, 2264535, 2333925, 2360085, 2365965, 2379465, 2465925, 2474955, 2499255, 2511495

Offset: 1

Zak Seidov, May 18 2006

Most terms are multiples of 5. In the first 130 terms, there only 19 non-multiples of 5: 2226609, 2556477, 3252249, 3496779, 3638439, 4060287, 4779621, 4821453, 5146713, 5313231, 5365899, 5504499, 5578419, 5738733, 5785857, 5845749, 6189183, 6222447, 6236769.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A086793, A119396-A119415.

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 by 2 while count < 100 do
  if f(n) = 21 then Res:= Res, n; count:= count+1 fi;
od:
Res; # Robert Israel, Apr 03 2018

a(1)=628425 inserted 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

A119397 Numbers taking exactly 21 steps to reach 15 in A086793.

Original entry on oeis.org

71820, 103950, 127764, 135660, 141360, 161460, 173250, 183744, 193284, 203580, 206712, 209440, 214830, 217620, 221760, 223020, 223860, 234432, 243540, 244440, 246330, 247752, 256680, 263160, 264180, 265200, 280368, 281160, 286380, 287712

Offset: 1

Zak Seidov, May 18 2006

Some trajectories are: 71820,1104,168,102,36,46,18,30,27,22,9,13,5,6,12,19,11,3,4,7,8,15 103950,1134,168,... 127764,924,168,... 135660,1134,168,... 221760,1908,162,66,36,... 343728,1650,162,... 376992,2016,297,66,36,... All trajectories eventually join one of previous trajectories. In the first 136 terms <800000 there is only one odd number, 628425. Next odd terms are: 824175, 1340325, 1422135, 1495725 (see A119398).

Cf. A086793, A119396-A119415.

A260059 Infinite square array whose n-th row lists the numbers k for which A086793(k)=n, where A086793 = number of iteration of A034690 (sum of digits of divisors) to reach a fixed point, read by antidiagonals.

Original entry on oeis.org

8, 14, 7, 20, 21, 4, 26, 39, 35, 3, 59, 43, 44, 54, 2, 62, 52, 48, 56, 11, 19, 122, 57, 49, 128, 101, 37, 12, 123, 61, 50, 171, 136, 73, 64, 6, 143, 67, 65, 182, 138, 109, 108, 29, 5, 149, 84, 99, 188, 160, 127, 301, 33, 23, 13, 167, 93, 104, 216, 184, 163, 553, 47, 24, 31, 9, 206, 112, 105, 248, 190, 181, 589, 83, 28, 38, 25, 10, 239

Offset: 1

M. F. Hasler, Nov 08 2015

The fixed points of A034690 are 1 and 15, these are the only numbers not appearing in this table. All other positive integers appear exactly once.
Is there a simple explanation why row 7 seems to grow significantly faster than the neighboring rows?
From row 21 on, the terms become very large: cf. A094501 which is the first column with 15 prefixed.

			The rows read
[ 8, 14,  20,  26,  59,  62, 122, 123, 143, 149, 167, 206, 239, 257, 293, 302,...],
[ 7, 21,  39,  43,  52,  57,  61,  67,  84,  93, 112, 124, 139, 151, 157, 189,...],
[ 4, 35,  44,  48,  49,  50,  65,  99, 104, 105, 116, 121, 125, 132, 140, 141,...],
[ 3, 54,  56, 128, 171, 182, 188, 216, 248, 252, 261, 264, 268, 270, 333, 387,...],
[ 2, 11, 101, 136, 138, 160, 184, 190, 208, 232, 238, 255, 282, 290, 318, 328,...],
[19, 37,  73, 109, 127, 163, 181, 271, 307, 396, 433, 523, 541, 613, 631, ...],
[12, 64, 108, 301, 553, 589, 949,1089,1197,1273,1687,1876,1957,2116, ...],
[ 6, 29,  33,  47,  83, 137, 173, 191, 227, 263, 281, 303, 317, ...],
[ 5, 23,  24,  28,  41,  42,  45,  92, 113, 131, 158, 164, ...],
[13, 31,  38,  60,  69,  74,  76,  77,  80,  86, 88, ...],
[ 9, 25,  72,  81, 117, 126, 156, 172, 258, 300, ...],
[10, 17,  22,  53,  71,  96, 107, 133, 202, ...], etc.
The first column is A094501.

Cf. A034690, A086793, A094501.

(f(k,N=20,a=[],n=0)=while(#aA086793(n++)==k&&a=concat(a,n));a); T=vector(20,n,f(n,21-n)); for(n=1,20,for(k=1,n,print1(T[k][n-k+1]",")))

A034690 Sum of digits of all the divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 9, 3, 19, 5, 15, 15, 22, 9, 30, 11, 15, 14, 9, 6, 33, 13, 15, 22, 29, 12, 27, 5, 27, 12, 18, 21, 46, 11, 24, 20, 27, 6, 33, 8, 21, 33, 18, 12, 52, 21, 21, 18, 26, 9, 48, 18, 48, 26, 27, 15, 42, 8, 15, 32, 37, 21, 36, 14, 36, 24, 36, 9, 69, 11, 24, 34

Offset: 1

Erich Friedman

For first occurrence of k, or 0 if k never appears, see A191000.
The only fixed points are 1 and 15. These are also the only loops of iterations of A034690: see the SeqFan thread "List the divisors...". - M. F. Hasler, Nov 08 2015
The following sequence is composed of numbers n such that the sum of digits of all divisors of n equals 15: 8, 14, 15, 20, 26, 59, 62, ... It actually depicts the positions of number 15 in this sequence: see the SeqFan thread "List the divisors...". - V.J. Pohjola, Nov 09 2015

			a(15) = 1 + 3 + 5 + (1+5) = 15. - _M. F. Hasler_, Nov 08 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Angelini et al., List the dividers [sic], sum the digits, lost messages reconstructed by _N. J. A. Sloane_, Dec 21 2024
H. Havermann et al, in reply to E. Angelini, List the dividers, sum the digits, SeqFan list, Nov. 2015. [Broken link]
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018-2020. See (22) p. 6.

Cf. A000005, A000203, A007953, A086793, A191000.

Cf. A093653 (binary equivalent)

a034690 = sum . map a007953 . a027750_row
-- Reinhard Zumkeller, Jan 20 2014

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;
# Alternative:
sd:= proc(n) option remember; local k; k:= n mod 10; k + procname((n-k)/10) end proc:
for n from 0 to 9 do sd(n):= n od:
a:= n -> add(sd(d), d=numtheory:-divisors(n)):
map(a, [$1..100]); # Robert Israel, Nov 17 2015

Table[Plus @@ Flatten@ IntegerDigits@ Divisors@n, {n, 75}] (* Robert G. Wilson v, Sep 30 2006 *)

vector(100, n, sumdiv(n, d, sumdigits(d))) \\ Michel Marcus, Jun 28 2015

A034690(n)=sumdiv(n,d,sumdigits(d)) \\ For use in other sequences. - M. F. Hasler, Nov 08 2015

from sympy import divisors
def sd(n): return sum(map(int, str(n)))
def a(n): return sum(sd(d) for d in divisors(n))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Oct 06 2021

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

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

cp's OEIS Frontend

A095347 n sets a new record for number of iterations of A034690 (sum of digits of the divisors of n) needed to reach 15 (see A086793).

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A114527 Numbers k such that A086793(k) is 1.

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Mathematica

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

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A119398 Odd numbers taking exactly 21 steps to reach 15 in A086793.

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Maple

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A118358 Records in A086793.

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A119397 Numbers taking exactly 21 steps to reach 15 in A086793.

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A260059 Infinite square array whose n-th row lists the numbers k for which A086793(k)=n, where A086793 = number of iteration of A034690 (sum of digits of divisors) to reach a fixed point, read by antidiagonals.

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PARI

A034690 Sum of digits of all the divisors of n.

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A106756 Primes with digit sum = 14.

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