A215942 a(n) = sigma(6*n) - 12*n.

0, 4, 3, 12, 12, 19, 12, 28, 12, 48, 12, 51, 12, 56, 54, 60, 12, 64, 12, 120, 60, 72, 12, 115, 72, 80, 39, 144, 12, 186, 12, 124, 72, 96, 156, 168, 12, 104, 78, 264, 12, 224, 12, 192, 180, 120, 12, 243, 96, 268, 90, 216, 12, 199, 204, 320, 96, 144, 12, 450

Offset: 1

Author

Michel Marcus, Aug 28 2012

Keywords

• nonn

Comments

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,

They are all probably A028982

(End)

Examples

			a(1) = sigma(6) - 2*6 = 12 - 12 = 0.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..1000

Programs

Mathematica

Table[DivisorSigma[1,6n]-12n,{n,60}] (* Harvey P. Dale, Aug 10 2024 *)

Formula

a(n) = sigma(6*n) - 12*n.