A191161 - OEIS

%I #23 Nov 22 2024 16:26:10

%S 1,4,5,12,7,22,9,32,19,30,13,72,15,38,37,80,19,90,21,96,47,54,25,208,

%T 39,62,65,120,31,178,33,192,67,78,65,316,39,86,77,272,43,222,45,168,

%U 147,102,49,560,67,174,97,192,55

%N Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

%C In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

%H Charles R Greathouse IV, <a href="/A191161/b191161.txt">Table of n, a(n) for n = 1..10000</a>

%H Jon Maiga, <a href="http://sequencedb.net/s/A191161">Computer-generated formulas for A191161</a>, Sequence Machine.

%F a(n) = sigma(n) + sum_{d | n, d < n} a(d). - _Charles R Greathouse IV_, Dec 20 2011

%F From _Antti Karttunen_, Nov 22 2024: (Start)

%F Following formulas were conjectured by Sequence Machine:

%F For n > 1, a(n) = A191150(n) + A074206(n).

%F a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.

%F a(n) = Sum_{d|n} A330575(d).

%F a(n) = Sum_{d|n} d*A067824(n/d).

%F a(n) = Sum_{d|n} A000203(d)*A074206(n/d).

%F a(n) = Sum_{d|n} A007429(d)*A174725(n/d).

%F a(n) = Sum_{d|n} A000010(d)*A253249(n/d).

%F a(n) = Sum_{d|n} A038040(d)*A323912(n/d).

%F a(n) = Sum_{d|n} A060640(d)*A323910(n/d).

%F (End)

%t hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)

%o (PARI) a(n)=sumdiv(n,d,if(d<n,d+a(d),n)) \\ _Charles R Greathouse IV_, Dec 20 2011

%Y Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).

%Y Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

%K nonn,easy

%O 1,2

%A _Alonso del Arte_, May 26 2011