A240265 -id:A240265

Search: a240265 -id:a240265

Displaying 1-5 of 5 results found.

page 1

A248915

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..8.

Michael S. Branicky, Python program for Andersen's algorithm extended to arbitrary base/ordering

StackExchange, New term in ascending order

EXAMPLE

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

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);

PROG

(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

CROSSREFS

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

KEYWORD

nonn,more,base

AUTHOR

Paolo P. Lava, Oct 16 2014

EXTENSIONS

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

STATUS

approved

A249125

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A084317, A069872, A224930, A240265.

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Oct 21 2014

STATUS

approved

A249764

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

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

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);

CROSSREFS

Cf. A066272, A069872, A224930, A240265.

KEYWORD

nonn,base

AUTHOR

Paolo P. Lava, Nov 05 2014

STATUS

approved

A308486

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

Similar to A308533 where anti-divisors are considered.

LINKS

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

EXAMPLE

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

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 *)

PROG

(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

(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

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Paolo P. Lava, May 31 2019

EXTENSIONS

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

STATUS

approved

A249765

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..8.

EXAMPLE

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

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);

CROSSREFS

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

KEYWORD

nonn,base,hard,more

AUTHOR

Paolo P. Lava, Nov 06 2014

EXTENSIONS

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

STATUS

approved

page 1

Search completed in 0.005 seconds