A235153 Let x(1)x(2)...x(q) the decimal expansion of the numbers k having exactly q distinct prime divisors p(1) < p(2) < ... < p(q). Sequence lists the numbers k such that p(1)/x(q) + p(2)/x(q-1)+ ... + p(q)/x(1) is an integer.

2, 3, 5, 7, 12, 24, 48, 132, 222, 234, 266, 364, 418, 468, 555, 663, 666, 2418, 2442, 3498, 4218, 4422, 6216, 6314, 6612, 8844, 21714, 26796, 28842, 41412, 61446, 62634, 66234, 82824, 491946, 641886, 648186, 6416718

Offset

1,1

Comments

The sequence is finite because the smallest number with 11 distinct divisors is k = 2*3*5*7*11*13*17*19*23*29*31 = 200560490130 with 12 decimal digits.

The corresponding integers are 1, 1, 1, 1, 4, 2, 1, 13, 21, 8, 11, 6, 16, 4, 9, 6, 7, 22, 23, 21, 22, 22, 13, 18, 12, 11, 39, 18, 17, 30, 17, 22, 22, 15, 30, 31, 25, 35.

Links

Table of n, a(n) for n=1..38.

Example

26796 is in the sequence because the five prime divisors are {2, 3, 7, 11, 29} and 2/6 + 3/9 + 7/7 + 11/6 + 29/2 = 18.

Maple

with(numtheory):
   for n from 1 to 1000000 do:
      x:=convert(n, base, 10):
      n1:=nops(x):
      p:=product('x[i]', 'i'=1..n1):
      y:=factorset(n):
      n2:=nops(y):
        if p<>0 and n1=n2
         then
         s:=sum('y[i]/x[i]', 'i'=1..n1):
          if s=floor(s)
           then
           printf(`%d, `, n):
           else
          fi:
        fi:
      od:

Prog

(PARI) is(k) = {my(d=digits(k), f=factor(k)[, 1], x); (x=#d)==#f && vecmin(d) && denominator(sum(i=1, x, f[i]/d[x-i+1]))==1; } \\ Jinyuan Wang, Mar 27 2020

Crossrefs

Cf. A235152.

Sequence in context: A086108 A052430 A344454 * A177968 A024784 A060528

Adjacent sequences: A235150 A235151 A235152 * A235154 A235155 A235156

Keyword

nonn,base,fini,full

Author

Michel Lagneau, Jan 04 2014

Extensions

a(38) from Jinyuan Wang, Mar 27 2020

Status

approved