A257491 Least k formed by the concatenation of two numbers n and d such that d is the n-th divisor of k, or 0 if no such k exists.

11, 22, 0, 410, 5125, 620, 735, 816, 975, 0, 0, 1280, 0, 14200, 0, 0, 17200, 1872, 19250, 2015625, 21109375, 222500, 0, 24150, 251250, 262000, 0, 0, 291450, 305000, 0, 0, 0, 0, 0, 0, 37115625, 0, 0, 402500, 0, 42336, 0, 442750, 45703125, 46920, 0, 4850000, 4912500, 0, 515100

Offset

1,1

Comments

Some semiprime near-misses for a(3): 35, 323, 365, 3103, 3173, 3493, 3755, 31673, 34993, 37495, 349993, 3166673, 34999993, 31666666673. - Michel Marcus, May 08 2015

Links

Robert Israel, Table of n, a(n) for n = 1..80

Example

a(4) = 410 because the divisors of 410 are {1, 2, 5, 10, 41, 82, 205, 410} and 10 is the 4th divisor of 410.

Maple

# this program tests k < 150000; results of 0 may be incorrect.
with(numtheory):nn:=51:
  for n from 1 to nn do:
   ii:=0:
    for k from n to 150000 while(ii=0)do:
     i:=length(k):p:=n*10^i+k:
      x:=divisors(p):n0:=nops(x):
      if n<=n0 then
      d:=x[n]:j:=length(d):q:=n*10^j+d:
      if p=q then
      ii:=1:printf ( "%d %d \n", n, p):
      else
      fi:fi:
    od:
    if ii=0 then printf ( "%d %d \n", n, 0):
    else
    fi:
   od:
# Alternative
f:= proc(n)
local F, m, t, primes, enum, cands, d, divs;
F:= ifactors(n)[2];
primes:= {seq(t[1], t=F)} union {2, 5};
for t in F do m[t[1]]:= t[2] od;
m[2]:= n; m[5]:= n;
enum:= proc(pr, t)
  local p, r;
  if pr = {} or t <= 1 then return [1] fi;
  p:= pr[1];
  r:= pr[2..-1];
  [seq(op(map(`*`, procname(r, floor(t/(1+j))), p^j)), j=0..m[p])]
end proc;
cands:= sort(enum(primes, n));
for d in cands do
  divs:= sort(convert(numtheory:-divisors(n*10^(1+ilog10(d))+d), list));
  if nops(divs) >= n and divs[n] = d then return(n*10^(1+ilog10(d))+d) fi;
od;
0
end proc:
seq(f(n), n=1..60); # Robert Israel, Jun 08 2015

Crossrefs

Cf. A000012 (1st divisor of n), A020639 (2nd divisor of n).

Sequence in context: A305660 A110393 A202627 * A169853 A064263 A138838

Adjacent sequences: A257488 A257489 A257490 * A257492 A257493 A257494

Keyword

nonn,base,hard

Author

Michel Lagneau, Apr 26 2015

Extensions

a(21), a(45) and a(48) from Robert Israel, Jun 08 2015

Status

approved