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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.
1
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.
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
KEYWORD
nonn,base,fini,full
AUTHOR
Michel Lagneau, Jan 04 2014
EXTENSIONS
a(38) from Jinyuan Wang, Mar 27 2020
STATUS
approved