A224930 -id:A224930 - cp's OEIS frontend

A069872 Numbers k such that k divides the concatenation all divisors in ascending order, i.e., k divides A037278(k).

1, 2, 4, 5, 6, 8, 10, 15, 16, 20, 24, 25, 30, 32, 40, 50, 60, 64, 80, 90, 96, 100, 104, 120, 124, 125, 128, 150, 160, 200, 240, 250, 255, 256, 288, 320, 360, 375, 380, 384, 400, 425, 464, 480, 495, 500, 512, 600, 618, 625, 640, 750, 795, 800, 864, 875, 960, 1000

Offset: 1

Amarnath Murthy, Apr 24 2002

All the powers of 2 are terms.

			16 is a term as 16 divides 124816, 24 is a term as 24 divides 1234681224.

Giovanni Resta, Table of n, a(n) for n = 1..1547 (terms <= 10^10, first 500 terms from Harvey P. Dale)

Cf. A037278, A224930.

k:=1; sol:=[];
for u in [1..1000] 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 u eq 0 then sol[k]:=u; k:=k+1; end if;
end for;
sol; // Marius A. Burtea, Jun 01 2019

Select[Range[1000],Divisible[FromDigits[Flatten[IntegerDigits/@ Divisors[ #]]],#]&] (* Harvey P. Dale, Dec 31 2012 *)

f(n) = my(d=divisors(n), s=""); fordiv(n, d, s = concat(s, Str(d))); eval(s); \\ A037278
isok(n) = f(n) % n == 0; \\ Michel Marcus, Jun 01 2019

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003

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

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

A379170 Let m be the concatenation, in descending order, of the divisors of k written in base 2 and then converted in base 10. Sequence lists k which divide m.

Original entry on oeis.org

1, 21, 253, 1407, 3425, 17457, 17459, 933661, 1279313, 29581875, 47960915, 76385733, 158292295, 3222873413, 3817850653, 319145363229

Offset: 1

Paolo P. Lava, Dec 17 2024

Tested up 4*10^11 by Giovanni Resta, Dec 22 2024

			Divisors of 21 are 1, 3, 7, 21 which in base 2 are 1, 11, 111, 10101. Their concatenation is 10101111111 which in base 10 is 1407. Finally 1407/21 = 67 is an integer, so 21 is a member of the sequence.

Cf. A224930, A379169.

with(numtheory): P:=proc(q) global a,b,c,k,n,v; v:=[];
for n from 1 to q do a:=sort([op(divisors(n))]); b:=0;
for k from 1 to nops(a) do c:=convert(a[-k],binary,decimal); b:=b*10^length(c)+c; od;
if frac(convert(b,decimal,binary)/n)=0 then v:=[op(v),n]; fi;
op(v); od; end: P(20000);

a(9)-a(16) from Giovanni Resta, Dec 22 2024

A249765 Numbers that divide the concatenation, in descending order, of their anti-divisors.

Original entry on oeis.org

7, 23957, 56483, 74651, 316782, 13594764, 14473747, 30056837

Offset: 1

Paolo P. Lava, Nov 06 2014

			Anti-divisors of 7 are and 2, 3, 5 and their concatenation in descending order is 532. Finally, 532 / 7 = 76.

Cf. A066272, A069872, A224930, A240265, A249764.

P:=proc(q) local a,k,n; for n from 3 to q do a:=0;
for k from n-1 by -1 to 2 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);

a(5)-a(8) from Chai Wah Wu, Nov 21 2014

cp's OEIS Frontend

A069872 Numbers k such that k divides the concatenation all divisors in ascending order, i.e., k divides A037278(k).

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

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

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

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A379170 Let m be the concatenation, in descending order, of the divisors of k written in base 2 and then converted in base 10. Sequence lists k which divide m.

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