A235152 -id:A235152

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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.

+10
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

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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:

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.

KEYWORD

nonn,base,fini,full

AUTHOR

Michel Lagneau, Jan 04 2014

EXTENSIONS

a(38) from Jinyuan Wang, Mar 27 2020

STATUS

approved

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