[seqfan] A215942

[michel.marcus at free.fr]

Hello SeqFans, 
  
I have recently entered a sequence : A215942(n) = sigma(6*n) -12*n because of a comment in A005101. 
Looking at A215942, I saw that very few n are such that A215942(n) is odd. 
For instance up to 100 : 3,6,12,24,27,48,54,75,96,... 
So I wondered what was this sequence when I realised that it seemed to be 3*A028982. 
  
Then I replaced (6,12) by other values (28,56), (12,28), (7,8), (120,360), ..., (i, sigma(i)), etc. 
Here is a summary of the results for i=1 to 10. 
  
sigma(i*n) - sigma(i)*n 
1:0,1,1,3,1,6,1,7,4,8, (sigma(n) -n: A001065) 
2:0,1,3,3,3,10,3,7,12,12, 
3:0,4,1,12,4,15,4,28,4,32, 
4:0,1,7,3,7,18,7,7,28,20, 
5:0,6,6,18,1,36,6,42,24,33, 
6:0,4,3,12,12,19,12,28,12,48, (A215942) 
7:0,8,8,24,8,48,1,56,32,64, 
8:0,1,15,3,15,34,15,7,60,36, 
9:0,13,1,39,13,42,13,91,4,104, 
10:0,6,18,18,3,60,18,42,72,37, 
they all begin with 0 obviously 
  
sequences of n such that the above is odd 
1:2,3,4,5,7,8,11,13,15,16, (sigma(n)-n is odd : A053868) 
2:2,3,4,5,7,8,11,13,15,16, 
3:3,6,12,24,27,48,54,75,96,108, 
4:2,3,4,5,7,8,11,13,15,16, 
5:5,10,20,40,45,80,90,125,160,180, 
6:3,6,12,24,27,48,54,75,96,108, 
7:7,14,28,56,63,112,126,175,224,252, 
8:2,3,4,5,7,8,11,13,15,16, 
9:2,3,4,5,7,8,11,13,15,16, 
10:5,10,20,40,45,80,90,125,160,180,  
the common factor of each line are : 1,1,3,1,5,3,7,1,1,5,11,3 

here we see that several lines are 2,3,4,5,7,8,11,13,15,16,  (probably A053868) 
they have index 1,2,4,8,9,16,18,25,32,... (probably A028982) 
and have a common factor 1 
  
the other lines have index 3,5,6,7,10,11,12,13,14,15, .. (probably A028983) and a common factor 3,5,3,7,5,11,3,13,7,15,17 
  
when different from A053868 each line divided by its common factor 
3:1,2,4,8,9,16,18,25,32,36, 
5:1,2,4,8,9,16,18,25,32,36, 
6:1,2,4,8,9,16,18,25,32,36, 
7:1,2,4,8,9,16,18,25,32,36, 
10:1,2,4,8,9,16,18,25,32,36,  
they are all probably A028982 
  
I don't know what to think about this. 
I hope there are no mistakes. Is this making any sense ?   

  
Cheers. 
Michel