%I #8 Mar 27 2020 09:32:12

%S 2,3,5,7,15,222,555,666,834,1122,2442,3162,4818,6162,6216,8274,8554,

%T 28842,49266,49434,61446,69762,83334,88638,88842,89838,641886,648186,

%U 795795,892164

%N 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(1) + p(2)/x(2) + ... + p(q)/x(q) is an integer.

%C 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.

%C The corresponding integers are 1, 1, 1, 1, 4, 21, 9, 7, 36, 19, 23, 22, 21, 45, 15, 52, 16, 19, 11, 33, 11, 79, 24, 18, 40, 13, 19, 26, 16, 31.

%e 28842 is in the sequence because the five prime divisors are {2, 3, 11, 19, 23} and 2/2 + 3/8 + 11/8 + 19/4 + 23/2 = 19.

%p with(numtheory):

%p for n from 1 to 1000000 do:

%p x:=convert(n,base,10):

%p n1:=nops(x):

%p p:=product('x[i]', 'i'=1..n1):

%p y:=factorset(n):

%p n2:=nops(y):

%p if p<>0 and n1=n2

%p then

%p s:=sum('y[i]/x[n1-i+1]', 'i'=1..n1):

%p if s=floor(s)

%p then

%p printf(`%d, `,n):

%p else

%p fi:

%p fi:

%p od:

%o (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[i]))==1; } \\ _Jinyuan Wang_, Mar 27 2020

%Y Cf. A235153.

%K nonn,base,fini,full

%O 1,1

%A _Michel Lagneau_, Jan 04 2014