A215942 - OEIS

%I #18 Aug 10 2024 14:27:44

%S 0,4,3,12,12,19,12,28,12,48,12,51,12,56,54,60,12,64,12,120,60,72,12,

%T 115,72,80,39,144,12,186,12,124,72,96,156,168,12,104,78,264,12,224,12,

%U 192,180,120,12,243,96,268,90,216,12,199,204,320,96,144,12,450

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

%C Motivated by the fact that sigma(6*n) >= 12*n.

%C If n is prime and n > 3, then a(n) = 12*(n+1) - 12*n = 12. - corrected by _Jonathan Sondow_, Sep 29 2012

%C 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

%C (Start)

%C I have recently entered A215942(n) = sigma(6*n) -12*n because of a comment in A005101.

%C Looking at A215942, I saw that there are very few n such that A215942(n) is odd.

%C For instance up to 100: 3,6,12,24,27,48,54,75,96,...  This appears to be 3*A028982.

%C Then I replaced (6,12) by other values (28,56), (12,28), (7,8), (120,360), ..., (i, sigma(i)), etc.

%C Here is a summary of the results for i=1 to 10.

%C sigma(i*n) - sigma(i)*n

%C 1: 0,1,1,3,1,6,1,7,4,8, (sigma(n) - n: A001065)

%C 2: 0,1,3,3,3,10,3,7,12,12,

%C 3: 0,4,1,12,4,15,4,28,4,32,

%C 4: 0,1,7,3,7,18,7,7,28,20,

%C 5: 0,6,6,18,1,36,6,42,24,33,

%C 6: 0,4,3,12,12,19,12,28,12,48, (A215942)

%C 7: 0,8,8,24,8,48,1,56,32,64,

%C 8: 0,1,15,3,15,34,15,7,60,36,

%C 9: 0,13,1,39,13,42,13,91,4,104,

%C 10: 0,6,18,18,3,60,18,42,72,37,

%C Values of n such that the above is odd:

%C 1: 2,3,4,5,7,8,11,13,15,16, (sigma(n) - n is odd: A053868)

%C 2: 2,3,4,5,7,8,11,13,15,16,

%C 3: 3,6,12,24,27,48,54,75,96,108,

%C 4: 2,3,4,5,7,8,11,13,15,16,

%C 5: 5,10,20,40,45,80,90,125,160,180,

%C 6: 3,6,12,24,27,48,54,75,96,108,

%C 7: 7,14,28,56,63,112,126,175,224,252,

%C 8: 2,3,4,5,7,8,11,13,15,16,

%C 9: 2,3,4,5,7,8,11,13,15,16,

%C 10: 5,10,20,40,45,80,90,125,160,180,

%C Gcd's of the above lines: 1,1,3,1,5,3,7,1,1,5,11,3

%C Several of these lines are 2,3,4,5,7,8,11,13,15,16, (probably A053868)

%C They have indices 1,2,4,8,9,16,18,25,32,... (probably A028982) and have a common factor 1

%C 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

%C When different from A053868 each line divided by its gcd gives:

%C 3: 1,2,4,8,9,16,18,25,32,36,

%C 5: 1,2,4,8,9,16,18,25,32,36,

%C 6: 1,2,4,8,9,16,18,25,32,36,

%C 7: 1,2,4,8,9,16,18,25,32,36,

%C 10: 1,2,4,8,9,16,18,25,32,36,

%C They are all probably A028982

%C (End)

%H Michel Marcus, <a href="/A215942/b215942.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%t Table[DivisorSigma[1,6n]-12n,{n,60}] (* _Harvey P. Dale_, Aug 10 2024 *)

%K nonn

%O 1,2

%A _Michel Marcus_, Aug 28 2012