A069872 -id:A069872 - cp's OEIS frontend

A224930 Numbers n such that n divides the concatenation of all divisors in descending order.

1, 561, 1703, 2883, 11623, 14721, 32431, 205119, 361767, 826471901, 3747204067, 17463443163, 404345080971, 573488405493, 5623233497397

Offset: 1

Paolo P. Lava, May 02 2013

Like A069872 but in descending order.
a(11) > 2*10^9. - Donovan Johnson, May 05 2013
a(16) > 10^13. - Giovanni Resta, Feb 14 2020

			Divisors of 561 are 1, 3, 11, 17, 33, 51, 187, 561 and 5611875133171131 / 561 = 10003342483371.
Divisors of 1703 are 1, 13, 131, 1703 and 1703131131 / 1703 = 1000077.

Cf. A069872.

with(numtheory); A224930:=proc(q) local a,b,c,d,f,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]); b:=nops(a); c:=a[b];
  for k from 1 to b-1 do d:=c; f:=0; while d>0 do f:=f+1; d:=trunc(d/10); od;
  c:=c+a[k+1]*10^f; od; if type(c/n,integer) then print(n); fi;
od; end: A224930 (10^6); # Paolo P. Lava, May 02 2013

Select[Range[10^6/2]*2-1, Mod[ FromDigits@ Flatten@ IntegerDigits[ Reverse@ Divisors@ #], #] == 0 &] (* Giovanni Resta, May 05 2013 *)

isok(k) = {my(s = "", d = Vecrev(divisors(k))); for (j=1, #d, s = concat(s, Str(d[j]));); eval(k) % k == 0;} \\ Michel Marcus, Feb 14 2020

a(10) from Donovan Johnson, May 05 2013
a(11)-a(12) from Giovanni Resta, May 05 2013
a(13)-a(14) from Giovanni Resta, May 10 2013
a(15) from Giovanni Resta, Feb 14 2020

A248915 Composite numbers which divide the concatenation of their prime factors, with multiplicity, in descending order.

Original entry on oeis.org

378, 12467, 95823, 10715274, 13485829, 111495095, 42002916561, 176685987695

Offset: 1

Paolo P. Lava, Oct 16 2014

Prime numbers are not considered because they trivially satisfy the relation.
For terms in ascending order see A259047 and StackExchange link. [Paolo P. Lava, May 30 2019]
a(9) <= 3953318131772867. - Chai Wah Wu, Apr 12 2024
a(2), the bound for a(9) above, and larger terms may be found using an extension of Andersen's algorithm to arbitrary base and ordering (see links for an implementation and another term). - Michael S. Branicky, Apr 13 2024

			Prime factors of 378 are 2,3,3,3,7; concat(7,3,3,3,2) = 73332 and 73332/378 = 194.

Michael S. Branicky, Python program for Andersen's algorithm extended to arbitrary base/ordering
StackExchange, New term in ascending order

Cf. A037276, A069872, A224930, A240265, A259047.

with(numtheory); P:=proc(q) local a,b,c,d,j,k,n;
for n from 1 to q do if not isprime(n) then a:=ifactors(n)[2]; b:=[]; d:=0;
for k from 1 to nops(a) do b:=[op(b),a[k][1]]; od; b:=sort(b);
for k from nops(a) by -1 to 1 do c:=1; while not b[k]=a[c][1] do c:=c+1; od;
for j from 1 to a[c][2] do d:=10^(ilog10(b[k])+1)*d+b[k]; od; od;
if type(d/n,integer) then print(n); fi;
fi; od; end: P(10^9);

isok(n) = {my(s = ""); my(f = factor(n)); forstep (i=#f~, 1, -1, for (k=1, f[i,2], s = concat(s, Str(f[i,1])))); (eval(s) % n) == 0;} \\ Michel Marcus, Jun 16 2015

a(7)-a(8) from Giovanni Resta, Jun 16 2015

A240265 Numbers that divide the concatenation of their aliquot divisors, in ascending order.

Original entry on oeis.org

1, 4, 15, 16, 255, 375, 495, 795, 1469, 3825, 9375, 28125, 66375, 67875, 234375, 249487, 286875, 309375, 337185, 450615, 590625, 628125, 1369125, 2390625, 2773125, 2781387, 3069375, 3706785, 4965309, 5859375, 12890625, 13539375, 26803125, 39607575, 62578125

Offset: 1

Paolo P. Lava, Apr 03 2014

The sequence is infinite, because it contains all the numbers of the form 3*5^(2k+1). - Giovanni Resta, Apr 03 2014

			Aliquot divisors of 1469 are 1, 13, 113. Their concatenation in ascending order is 113113 and 113113/1469 = 77.

Cf. A069872, A224930.

with(numtheory);
T:=proc(t) local x,y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
P:=proc(q) local a,b,c,d,i,k,n;
for n from 2 to q do a:=sort([op(divisors(n))]); b:=a[nops(a)-1];
for i from nops(a)-2 by -1 to 1 do b:=b+a[i]*10^T(b); od;
if type(b/n,integer) then print(n); fi;
od; end: P(10^6);

Select[Range[6258*10^4],Divisible[FromDigits[Flatten[IntegerDigits/@ Most[ Divisors[ #]]]],#]&] (* Harvey P. Dale, Aug 21 2019 *)

a(14)-a(34) from Giovanni Resta, Apr 03 2014

First term (a(1) = 1) prepended by Harvey P. Dale, Aug 21 2019

A077352 a(n) = (concatenation in ascending order of divisors of 2^n)/2^n.

Original entry on oeis.org

1, 6, 31, 156, 7801, 390051, 19502551, 9751275501, 4875637750501, 2437818875250501, 12189094376252505001, 60945471881262525005001, 304727359406312625025005001, 1523636797031563125125025005001, 76181839851578156256251250250050001, 3809091992578907812812562512502500050001

Offset: 0

Amarnath Murthy, Nov 05 2002

			a(6) = 1248163264/64 = 19502551.

Paolo Xausa, Table of n, a(n) for n = 0..80

Cf. A000079, A034887, A059268, A077351, A069872, A077353.

a:= n-> parse(cat(2^i$i=0..n))/2^n:
seq(a(n), n=0..15);  # Alois P. Heinz, May 16 2025

A077352[n_] := FromDigits[Flatten[IntegerDigits[Divisors[#]]]]/# & [2^n];
Array[A077352, 16, 0] (* or *)
FoldList[10^IntegerLength[2^#2]*#/2 + 1 &, 1, Range[15]] (* Paolo Xausa, May 19 2025 *)

For n>=1, a(n) = (a(n-1)*2^(n-1)*10^(floor(log_10(2^n))+1)+2^n)/2^n. - Sam Alexander, Feb 27 2005

Offset corrected by Sean A. Irvine, May 16 2025

A077353 a(n) = (concatenation in ascending order of divisors of 5^n)/5^n.

Original entry on oeis.org

1, 3, 61, 12201, 2440201, 4880402001, 97608040020001, 1952160800400020001, 390432160080004000200001, 780864320160008000400002000001, 1561728640320016000800004000002000001, 31234572806400320016000080000040000020000001, 6246914561280064003200016000008000004000000200000001

Offset: 0

Amarnath Murthy, Nov 05 2002

			a(5) = 15251256253125/3125 = 4880402001.

Paolo Xausa, Table of n, a(n) for n = 0..50

Cf. A000351, A055642, A077351, A069872, A077352.

a:= n-> parse(cat(5^i$i=0..n))/5^n:
seq(a(n), n=0..12);  # Alois P. Heinz, May 16 2025

A077353[n_] := FromDigits[Flatten[IntegerDigits[Divisors[#]]]]/# & [5^n];
Array[A077353, 16, 0] (* or *)
FoldList[10^IntegerLength[5^#2]/5*# + 1 &, 1, Range[15]] (* Paolo Xausa, May 19 2025 *)

a(n) = eval(concat(apply(x->Str(x),divisors(5^n))))/5^n \\ Max Alekseyev, Dec 12 2011

a(n) = if(n==0,1,(10^#Str(5^n)/5)*a(n-1)+1) \\ Jason Yuen, Aug 21 2024

a(0) = 1, a(n) = (10^A055642(5^n)/5)*a(n-1) + 1. - Jason Yuen, Aug 21 2024

More terms from Max Alekseyev, Dec 12 2011

A078218 Smallest multiple of n that begins with the concatenation of the divisors of n (in increasing order).

Original entry on oeis.org

1, 12, 132, 124, 15, 1236, 175, 1248, 1395, 12510, 1111, 12346128, 1131, 127148, 13515, 124816, 1173, 12369186, 1197, 12451020, 137214, 12112210, 12305, 1234681224, 1525, 1213264, 1392714, 1247142820, 12905, 12356101530, 13113

Offset: 1

Amarnath Murthy, Nov 22 2002

			The concatenation of the divisors of 7 is 17; 175 = 25*7 is the smallest multiple of 7 that begins with 17, so a(7) = 175.

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

Cf. A037278, A069872, A078810.

cdiv:= proc(n) local D,R,j;
  D:= sort(convert(numtheory:-divisors(n),list));
  R:= D[1];
  for j from 2 to nops(D) do
    R:= R * 10^(1+ilog10(D[j])) + D[j];
  od;
  R
end proc:
f:= proc(n) local t,i,r;
  t:= cdiv(n);
  for i from 0 do
    r:= n * ceil(t*10^i/n);
    if r < (t+1)*10^i then return r fi
  od
end proc:
map(f, [$1..50]); # Robert Israel, Oct 07 2024

{for(n=1,31,k=floor(log(n)/log(10))+1; d=divisors(n); v=Str(); for(i=1,matsize(d)[2], v=concat(v,Str(d[i]))); s=eval(v); t=s+1; m=floor(log(s)/log(10))+1; d=k-m; s=s*10^d; t=t*10^d; b=1; while(b>0,q=floor(s/n); while(b>0&&(p=q*n)=s,print1(p,","); b=0,q++)); s=10*s; t=10*t))}

Edited and extended by Klaus Brockhaus, Dec 06 2002

A249125 Composite numbers which are a multiple of the concatenation of their prime factors A084317.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 50, 64, 81, 100, 121, 125, 128, 169, 200, 243, 250, 256, 289, 343, 361, 400, 500, 512, 529, 625, 729, 800, 841, 961, 1000, 1024, 1250, 1331, 1369, 1600, 1681, 1849, 2000, 2048, 2187, 2197, 2209, 2401, 2500, 2809, 3125, 3200, 3481, 3721, 4000, 4096, 4489, 4913, 5000, 5041, 5329, 6241, 6250, 6400

Offset: 1

M. F. Hasler, Oct 21 2014

Prime numbers are excluded since they trivially satisfy the condition.
Multiplicity of the prime factors is ignored.
Among the first 10000 terms, the 182 which are not prime powers are of the form 2^h * 5^k. - Giovanni Resta, May 29 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A084317, A069872, A224930, A240265.

Select[Range[6400], CompositeQ[#] &&  Mod[#, FromDigits@ Flatten[ IntegerDigits /@ First /@ FactorInteger@#]] == 0 &] (* Giovanni Resta, May 29 2017 *)

for(n=2,9999,isprime(n)||n%A084317(n)||print1(n","))

A249764 Numbers which divide the concatenation, in ascending order, of their anti-divisors.

Original entry on oeis.org

15, 30, 105, 120, 150, 222, 375, 585, 1500, 1695, 1755, 1800, 2700, 3449, 3750, 3840, 4891, 6720, 7680, 12000, 13583, 14400, 15000, 18750, 19200, 20940, 28134, 30000, 34800, 35625, 46875, 48000, 68400, 72504, 75000, 93750, 120000, 128400

Offset: 1

Paolo P. Lava, Nov 05 2014

			Anti-divisors of 15 are 2, 6, 10 and their concatenation in ascending order is 2610. Finally, 2610 / 15 = 174.

Chai Wah Wu, Table of n, a(n) for n = 1..103

Cf. A066272, A069872, A224930, A240265.

P:=proc(q) local a,k,n; for n from 3 to q do a:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a*10^(ilog10(k)+1)+k; fi; od;
if type(a/n,integer) then print(n); fi; od; end: P(10^9);

A308486 Numbers such that the sum of divisors divides the concatenation (in ascending order) of divisors.

Original entry on oeis.org

1, 2, 6, 10, 40, 98, 112, 120, 1904, 2680, 4040, 4128, 5136, 9920, 12224, 17900, 20880, 27800, 44160, 55520, 57121, 62240, 86866, 158880, 178120, 1431808, 1773920, 1825280, 1918640, 3751328, 5452288, 6749600, 7262120, 7446720, 9916832, 17777440, 46168000, 101829808

Offset: 1

Paolo P. Lava, May 31 2019

Numbers k such that A000203(k) divides A037278(k). - Michel Marcus, Jun 02 2019.

Similar to A308533 where anti-divisors are considered.

			Divisors of 98 are 1, 2, 7, 14, 49, 98 and their sum is sigma(98) = 171. Then, 127144998 / 171 = 743538.

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A000203, A037278, A069872, A224930, A240265, A308533.

k:=1; sol:=[];
for u in [1..10000000] do D:=Divisors(u); conc:=D[1];
    for u1 in [2..#D] do a:=#Intseq(conc); a1:=#Intseq(D[u1]); conc:=10^a1*conc+D[u1];
    end for;
      if conc mod SumOfDivisors(u) eq 0 then sol[k]:=u; k:=k+1; end if;
end for;
sol; // Marius A. Burtea, Jun 01 2019

with(numtheory): P:=proc(q) local n; for n from 1 to q do if frac(parse(cat(op(sort([op(divisors(n))]))))/sigma(n))=0 then
print(n); fi; od; end: P(10^6);

Select[Range[10^6], Mod[FromDigits@ Flatten@ IntegerDigits[#], Total@ #] == 0 &@ Divisors@ # &] (* Michael De Vlieger, Jun 03 2019 *)

concd(n) = my(d=divisors(n), s=""); fordiv(n, d, s = concat(s, Str(d))); eval(s); \\ A037278
isok(n) = (concd(n) % sigma(n)) == 0; \\ Michel Marcus, Jun 05 2019

a(30)-a(38) from Giovanni Resta, May 31 2019

A357692 Integers k such that A037278(k) is a term of A175252.

Original entry on oeis.org

1, 2, 4, 15, 16, 25, 60, 90, 100, 124, 150, 240, 375, 384, 600, 618, 625, 960, 1536, 3330, 3750, 4650, 5760, 10000, 10500, 10752, 15000, 16384, 17500, 24576, 25600, 40000, 49500, 62500, 102400, 139200, 168750, 198400, 323280, 526848, 960000, 1179648, 1248000, 1369125

Offset: 1

Michel Marcus, Oct 10 2022

			A037278(4) = 124, a term of A175252.
A037278(15) = 13515, a term of A175252.
A037278(16) = 124816, a term of A175252.
A037278(25) = 1525, a term of A175252.

David A. Corneth, Table of n, a(n) for n = 1..81

Subsequence of A069872.

Cf. A037278, A175252.

is(n, {u = 10^5}) = {my(e = eval(concat(concat([""], divisors(n))))); if(e % n != 0, return(0); ); my(oldu = u, s, d); u = min(e, u); s = ""; d = divisors(factor(e, u)); d = select(x -> x <= u, d); for(i = 1, #d, s=concat(s, Str(d[i])); if(eval(s) == e, return(1)); if(eval(s) > e, return(0)); ); is(n, 10*oldu); } \\ David A. Corneth, Oct 10 2022, adapted from Michel Marcus' isok at A175252

More terms from David A. Corneth, Oct 10 2022

cp's OEIS Frontend

A224930 Numbers n such that n divides the concatenation of all divisors in descending order.

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A248915 Composite numbers which divide the concatenation of their prime factors, with multiplicity, in descending order.

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A240265 Numbers that divide the concatenation of their aliquot divisors, in ascending order.

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A077352 a(n) = (concatenation in ascending order of divisors of 2^n)/2^n.

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A077353 a(n) = (concatenation in ascending order of divisors of 5^n)/5^n.

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A078218 Smallest multiple of n that begins with the concatenation of the divisors of n (in increasing order).

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A249125 Composite numbers which are a multiple of the concatenation of their prime factors A084317.

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