OFFSET
1,1
COMMENTS
Let D0 = {d0(i)}, i = 1..p, the set of the p even divisors of a number n and D1 = {d1(n)}, j = 1..q the set of the q odd divisors of n. Then a(n) is the number such that 3*Sum_{i=1..p} 1/d0(i)- 2*Sum_{j=1..q} 1/d1(j) = 1.
Property of the sequence:
We observe that a(n) = 2^(k+1)*(2^k-1)*(2^(k+1) - 3) = (2*A000668(m) + 2)*A000668(m)*(2*A000668(m) - 1) where A000668(m) = 2^k - 1 is a Mersenne prime and (2*A000668(m)-1) = 2^(k+1)- 3 is also a prime number.
The corresponding values of k are 2, 3, 5, 13, 19, ... and the corresponding values of m are 1, 2, 3, 5, 7, ...
Generalization:
It is possible to introduce general sequences of numbers such that a*s0 + b*s1 = c with very interesting properties for some integers a, b, c.
Example 1: with (a, b, c) = (2, -1, 1) we find the sequence A064591 = 24, 112, 1984, 32512, ... (non-unitary perfect numbers).
Example 2: with (a, b, c) = (2, -1, 0) we find the sequence A016825(n) = 2, 6, 10, 14, 18, 22, ...
Example 3: with (a, b, c) = (1, 1, 2) we find the sequence A000396(n) = 6, 28, 496, 8128,... (perfect numbers).
Example 4: with (a, b, c) = (4, -3, 1) we find the sequence 48, 224, 3968, 65024, ... = 2*A064591(n) = A000668(n)*2^p for some p where A000668 lists the Mersenne primes.
Example 5: with (a, b, c) = (6, -5, 1) we find the sequence 240, 2912, 242048, ... which equals twice the sequence obtained with (a, b, c) = (3, -2, 1).
Example 6: with (a, b, c) = (7, -6, 1) we find the sequence 2150, 13104, 24800, ...
EXAMPLE
MAPLE
with(numtheory):nn:=100000:
for n from 2 by 2 to nn do :
x:=divisors(n):n0:=nops(x):s:=sum('x[i]', 'i'=1..n0):
s0:=0:s1:=0:
for k from 1 to n0 do:
if irem(x[k], 2)=0
then
s0:=s0+1/x[k]
else
s1:=s1+1/x[k]:
fi:
od:
if 3*s0-2*s1=1 then print(n):else fi:od:
MATHEMATICA
Do[s0=0; s1=0; Do[d=Divisors[n][[i]]; If[Mod[d, 2]==0, s0=s0+1/d, s1=s1+1/d], {i, 1, Length[Divisors[n]]}]; If[3*s0-2*s1==1, Print[n]], {n, 2, 10^9, 2}]
PROG
(PARI) siod(n) = sumdiv(n, d, (d%2)/d);
seod(n) = sumdiv(n, d, (1-d%2)/d);
isok(n) = 3*seod(n)-2*siod(n) == 1; \\ Michel Marcus, May 16 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, May 16 2015
STATUS
approved