A235602 -id:A235602 - cp's OEIS frontend

A235600 a(n) = n/d(n) if d(n) divides n, otherwise a(n) = n, where d(n) is the sum of the digits of n (A007953).

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 4, 13, 14, 15, 16, 17, 2, 19, 10, 7, 22, 23, 4, 25, 26, 3, 28, 29, 10, 31, 32, 33, 34, 35, 4, 37, 38, 39, 10, 41, 7, 43, 44, 5, 46, 47, 4, 49, 10, 51, 52, 53, 6, 55, 56, 57, 58, 59, 10, 61, 62, 7, 64, 65, 66, 67, 68, 69, 10, 71, 8, 73, 74, 75, 76, 77, 78, 79, 10, 9, 82, 83, 7, 85

Offset: 1

N. J. A. Sloane, Jan 18 2014

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A007953, A235601, A235602.

Cf. A065517 (first differs at n=15).

A235600[n_]:=With[{d=Total[IntegerDigits[n]]},If[Divisible[n,d],n/d,n]];
Array[A235600,100] (* Paolo Xausa, Dec 06 2023 *)

a(n) = my(s=sumdigits(n)); if (n % s, n, n/s); \\ Michel Marcus, Jul 15 2021

A235601 Smallest number m such that repeated application of A235600 takes n steps to reach 1, where A235600(k) = k/A007953(k) if the digital sum A007953(k) divides k, A235600(k) = k otherwise.

Original entry on oeis.org

1, 2, 12, 108, 1944, 52488, 1102248, 44641044, 2008846980, 108477736920, 6508664215200, 421761441144960, 22142475660110400, 1793540528468942400, 160701231350817239040, 15909421903730906664960, 1874419162475932276162560

Offset: 0

N. J. A. Sloane and David W. Wilson, Jan 18 2014

Numbers m > 1 which never reach 1 are not candidates for a(n).
There is no analog in base 2 (cf. A235602).
Comment from David W. Wilson, Jan 20 2013: let S(0) = {1};  for each n >= 1, compute the set S(n) of possible predecessors of elements of S(n-1).  Then a(n) is the smallest element of S(n). Using this approach, I was able to compute up to a(100).
The sequence is finite with a(440), a 1434-digit number being the final term. - Hans Havermann and Ray Chandler, Jan 21 2014
Sequence A236338 gives the count of iterations of A235600 required to reach 1 when starting from any n. Otherwise said: This sequence is the RECORDS transform of A236338. - M. F. Hasler, Jan 22 2014
The terms are a proper subset of A114440. - Robert G. Wilson v, Jan 22 2014

			a(4) = 1944: 1944 ->1944/18 = 108 -> 108/9 = 12 -> 12/3 = 4 -> 4/4 = 1 in 4 steps.

Hans Havermann and Ray Chandler, Table of n, a(n) for n = 0..440 [First 100 terms were computed by David W. Wilson]
David W. Wilson, Ray Chandler, Alonso Del Arte, M. F. Hasler, Hans Havermann, Alex Meiburg, N. J. A. Sloane, Hugo Van Der Sanden, and Allan Wechsler, As much as I hate "base" sequences..., Copies of various posts to the Sequence Fans Mailing List, Circa January 2014. Assembled by _N. J. A. Sloane_, Dec 23 2024

Cf. A005349, A007953, A114440, A235600, A236385.

s={1}; Print[s[[1]]]; Do[t={}; Do[v=s[[k]]; u={}; Do[If[Total[IntegerDigits[c*v]]==c, AppendTo[u,c*v]], {c,2,7000}]; t=Join[t,u], {k,Length[s]}]; s=Sort[t]; Print[s[[1]]], {440}] (* Hans Havermann, Jan 21 2014 *)

a(8) from Hans Havermann, Jan 19 2014
a(9)-a(100) from David W. Wilson, Jan 21 2014
a(101)-a(440) from Hans Havermann and Ray Chandler, Jan 21 2014

A358139 Numbers k > 0 sorted by k/A000120(k) in increasing order. A000120 is the binary weight of k. If k/A000120(k) yields equal values, the smaller k will appear first.

Original entry on oeis.org

1, 3, 2, 7, 5, 6, 11, 15, 4, 13, 9, 14, 10, 23, 12, 31, 19, 27, 21, 29, 22, 30, 8, 25, 17, 26, 18, 28, 47, 39, 20, 63, 43, 55, 45, 46, 35, 59, 24, 61, 37, 62, 38, 51, 53, 54, 41, 42, 57, 58, 44, 60, 79, 95, 16, 49, 33, 50, 34, 52, 87, 71, 36, 127, 91, 111, 93, 56

Offset: 1

Thomas Scheuerle, Oct 31 2022

A permutation of the positive integers.

This permutation satisfies a weak ordering: If b = a(c*d) and e = a(c) and f = a(d) then b > e and b > f with c,d > 1.

Thomas Scheuerle, a(1..25000) as scatter plot.
Thomas Scheuerle, a(1..1000) as scatter plot colored by binary weight and a(n)/A000120(a(n)) plotted as blue line.
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A049445, A161764, A199238, A235602, A348416.

f(x) = x/hammingweight(x);
cmpb(x, y) = my(hx=f(x), hy=f(y)); if (hx != hy, return(sign(hx-hy))); return(sign(x-y));
lista(nn) = Vec(vecsort([1..2*nn], cmpb, 1), nn); \\ Michel Marcus, Nov 05 2022

a(2^n) = 2^(n+1) - 1.

abs(a(n)-n) < n.

cp's OEIS Frontend

A235600 a(n) = n/d(n) if d(n) divides n, otherwise a(n) = n, where d(n) is the sum of the digits of n (A007953).

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A235601 Smallest number m such that repeated application of A235600 takes n steps to reach 1, where A235600(k) = k/A007953(k) if the digital sum A007953(k) divides k, A235600(k) = k otherwise.

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A358139 Numbers k > 0 sorted by k/A000120(k) in increasing order. A000120 is the binary weight of k. If k/A000120(k) yields equal values, the smaller k will appear first.

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