cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Paolo P. Lava, Oct 16 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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);
  • PARI
    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

Extensions

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

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

Views

Author

M. F. Hasler, Oct 21 2014

Keywords

Comments

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

Links

Crossrefs

Cf. A084317, A069872, A224930, A240265.

Programs

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

Views

Author

Paolo P. Lava, Nov 05 2014

Keywords

Examples

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

Links

Crossrefs

Cf. A066272, A069872, A224930, A240265.

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, May 31 2019

Keywords

Comments

Numbers k such that A000203(k) divides A037278(k). - Michel Marcus, Jun 02 2019.
Similar to A308533 where anti-divisors are considered.

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

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

Views

Author

Paolo P. Lava, Nov 06 2014

Keywords

Examples

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

Crossrefs

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

Programs

  • Maple
    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);

Extensions

a(5)-a(8) from Chai Wah Wu, Nov 21 2014
Showing 1-5 of 5 results.

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