maybe evident, maybe cool?

Meeussen Wouter (bkarnd) wouter.meeussen at vandemoortele.com
Wed Nov 28 18:44:28 CET 2001

```or, about a hit by superseeker:

if you apply GCD to all k-subsets of n,
and sum over the different subsets, you get:

Table[   Plus@@ ( GCD @@@ KSubsets[Range[n],m]) ,
{n,16},{m, n}]//ColumnForm

{1}
{3,1}
{6,3,1}
{10,7,4,1}
{15,11,10,5,1}
{21,20,21,15,6,1}
{28,26,36,35,21,7,1}
{36,38,60,71,56,28,8,1}
{45,50,90,127,126,84,36,9,1}
{55,67,132,215,253,210,120,45,10,1}
{66,77,177,335,463,462,330,165,55,11,1}
{78,105,250,512,798,925,792,495,220,66,12,1}
{91,117,316,732,1293,1717,1716,1287,715,286,78,13,1}
{105,142,409,1038,2023,3010,3433,3003,2002,1001,364,91,14,1}
{120,172,516,1410,3026,5012,6436,6435,5005,3003,1365,455,105,15,1}
{136,204,648,1902,4426,8036,11448,12871,11440,8008,4368,1820,560,120,16,1}

example:
a(4,2) = 7 since
GCD[1,2]+ GCD[1,3]+ GCD[1,4]+ GCD[2,3]+ GCD[2,4]+ GCD[3,4] = 7

The surprise (?) is in the row-sums=
1,4,10,22,42,84,154,298,568,1108,2142,4254,8362,16636,33076,66004

SuperSeeker found that the first differences of them equal A034738:
%N A034738 Dirichlet convolution of b_n=2^(n-1) with phi(n).
%F A034738 (1/2)* Sum_{d|n} phi(d)*2^(n/d), n >= 1.

{1}
{2, 1}
{4, 2}
{8, 2, 2}
{16, 4}
{32, 4, 4, 2}
{64, 6}
{128, 8, 4, 4}
{256, 8, 6}
{512, 16, 8, 4}
{1024, 10}
{2048, 32, 16, 8, 4, 4}
{4096, 12}
{8192, 64, 12, 6}
{16384, 32, 16, 8}
{32768, 128, 16, 8, 8}

is this evident, or is it cool?

Wouter Meeussen
tel  +32 (0)51 332 124
fax +32 (0)51 332 175
mail: wouter.meeussen at vandemoortele.com

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