A278441 -id:A278441 - cp's OEIS frontend

A278440 Numbers n such that A244112(n) | n.

22, 777, 4444, 68868, 200000, 303030, 333000, 333333, 555555, 660000, 660660, 666666, 700000, 2332200, 3131313, 4444400, 6060600, 7007000, 7700000, 9009790, 9656955, 9885585, 11517771, 14233221, 14331231, 14333110, 14411040, 15143331, 15233221, 15331231, 15333110

Offset: 1

Paolo P. Lava, Nov 25 2016

A244112(68868) = 3826 and 68868 / 3826 = 18.

Paolo P. Lava, Table of n, a(n) for n = 1..200

Cf. A237605, A244112, A278439, A278441.

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n; for n from 1 to q do a:=sort(convert(n,base,10));
for k from 1 to trunc(nops(a)/2) do c:=a[k]; a[k]:=a[nops(a)-k+1]; a[nops(a)-k+1]:=c; od;  k:=1; b:=a[1]; c:=0;
for j from 2 to nops(a) do if a[j]=b then k:=k+1; else d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; k:=1; b:=a[j]; fi; od;
d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; if type(n/c,integer) then print(n); fi; od; end: P(10^99);

Select[Range[10^6], Divisible[#, FromDigits@ Flatten@ Map[IntegerDigits, DeleteCases[#, k_ /; First@ k == 0]] &@ Reverse@ MapIndexed[{#1, (First@ #2 - 1)} &, RotateRight@ DigitCount@ #]] &] (* Michael De Vlieger, Dec 12 2016 *)

A278439 Numbers k such that k | A047842(k).

Original entry on oeis.org

1, 2, 5, 22, 105, 188, 258, 327, 663, 15425, 16654, 27848, 40324, 80328, 481263, 10213223, 10311233, 10313314, 10313315, 10313316, 10313317, 10313318, 10313319, 21322314, 21322315, 21322316, 21322317, 21322318, 21322319, 31123314, 31123315, 31123316, 31123317

Offset: 1

Paolo P. Lava, Nov 22 2016

The sequence is bounded. Let us consider a k-digit number n in which all 10 numerals from 0 to 9 are equally distributed: there are k/10 0's, k/10 1's, etc. This is the best case in order to have a number with the greatest number of digits under the transform n -> A047842(n). The number of digits we get is 10 + 10*floor(log_10(k/10) + 1), which must be >= k. The inequality becomes log_10(k/10) >= k/10 - 2, which is solved by k <= 23.75... This means that no term of the sequence can have more than 23 digits.

			A237605(258) = 121518 and 121518/258 = 471.

Paolo P. Lava, List of terms and corresponding ratios for n <= 10^9.

Cf. A047842, A237605, A278440, A278441.

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n; for n from 1 to q do
a:=sort(convert(n,base,10)); k:=1; b:=a[1]; c:=0; for j from 2 to nops(a) do
if a[j]=b then k:=k+1; else d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; k:=1; b:=a[j]; fi; od;
d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; if type(c/n,integer) then print(n); fi; od; end: P(10^10);

a(32) corrected by Sean A. Irvine, May 27 2025

A343455 Numbers where A244112(n) = n.

Original entry on oeis.org

22, 14233221, 14331231, 14333110, 15143331, 15233221, 15331231, 15333110, 16143331, 16153331, 16233221, 16331231, 16333110, 17143331, 17153331, 17163331, 17233221, 17331231, 17333110, 18143331, 18153331, 18163331, 18173331, 18233221, 18331231, 18333110, 19143331

Offset: 1

J. Lowell, Apr 15 2021

Intersection of A278440 and A278441.

Fixed points of A244112.

			14233221 has one 4, two 3's, three 2's, and two 1's.

Martin Ehrenstein, Table of n, a(n) for n = 1..57

Cf. A244112, A278440, A278441.

a(9) and beyond from Martin Ehrenstein, Apr 16 2021

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A278440 Numbers n such that A244112(n) | n.

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A278439 Numbers k such that k | A047842(k).

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A343455 Numbers where A244112(n) = n.

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