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COMMENTS
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Motivated by the fact that sigma(6*n) >= 12*n.
If n is prime and n > 3, then a(n) = 12*(n+1) - 12*n = 12. - corrected by Jonathan Sondow, Sep 29 2012
Michel Marcus posted the following comments about this sequence to the Sequence Fans Mailing List, and I think they are interesting enough to be included here - N. J. A. Sloane, Aug 30 2012
(Start)
I have recently entered A215942(n) = sigma(6*n) -12*n because of a comment in A005101.
Looking at A215942, I saw that there are very few n such that A215942(n) is odd.
For instance up to 100: 3,6,12,24,27,48,54,75,96,... This appears 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,
Values 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,
Gcd's of the above lines: 1,1,3,1,5,3,7,1,1,5,11,3
Several of these lines are 2,3,4,5,7,8,11,13,15,16, (probably A053868)
They have indices 1,2,4,8,9,16,18,25,32,... (probably A028982) and have a common factor 1
The other lines have indices 3,5,6,7,10,11,12,13,14,15, .. (probably A028983) and gcd's 3,5,3,7,5,11,3,13,7,15,17
When different from A053868 each line divided by its gcd gives:
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,
(End)
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