A257996 Let s0 and s1 be the sums of the reciprocals of the even and odd divisors of n, respectively. The sequence lists the numbers n such that 3*s0 - 2*s1 = 1.

120, 1456, 121024, 2198352216064, 576458003527499776

Offset: 1

Author

Michel Lagneau, May 16 2015

Keywords

• nonn

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, ...

Examples

			120 = 2^3*3*5 = (2*A000668(1)+2)* A000668(1)*(2*A000668(1)-1);
1456 = 2^4*7*13 = (2*A000668(2)+2)* A000668(2)*(2*A000668(2)-1);
121024 = 2^6*31*61 =(2*A000668(3)+2)* A000668(3)*(2*A000668(3)-1);
2198352216064 = 2^14*8191*16381= (2*A000668(5)+2)*A000668(5)*(2*A000668(5)-1);
576458003527499776 = 2^20*524287*1048573 = (2*A000668(7)+2)* A000668(7)*(2*A000668(7)-1).
		

Crossrefs

Cf. A000668.

Programs

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}]

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