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A191150 Hypersigma(n), definition 1: sum of the divisors of n plus the recursive sum of the divisors of the restricted divisors.

%I A191150 #43 Nov 22 2024 13:31:43
%S A191150 1,3,4,10,6,19,8,28,17,27,12,64,14,35,34,72,18,82,20,88,44,51,24,188,
%T A191150 37,59,61,112,30,165,32,176,64,75,62,290,38,83,74,252,42,209,44,160,
%U A191150 139,99,48,512,65,166,94,184,54,306,90,316,104,123,60,588,62,131
%N A191150 Hypersigma(n), definition 1: sum of the divisors of n plus the recursive sum of the divisors of the restricted divisors.
%C A191150 First we add up all the divisors of n, and then we add in the divisors of each restricted divisor of n (not 1 or n itself) and continue the recursion until such a depth as that there only numbers with no restricted divisors (prime numbers).
%C A191150 Thus if n is prime then hypersigma(n) is the same as sigma(n).
%H A191150 Alois P. Heinz, <a href="/A191150/b191150.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Alonso del Arte)
%H A191150 Jon Maiga, <a href="http://sequencedb.net/s/A191150">Computer-generated formulas for A191150</a>, Sequence Machine.
%F A191150 a(n) = n + 1 <=> n is prime. - _Bill McEachen_, Aug 01 2023
%F A191150 For n > 1, a(n) = A191161(n) - A074206(n). [Conjectured by Sequence Machine] - _Antti Karttunen_, Nov 22 2024
%e A191150 a(12) = 64 since: the sum of the divisors of 12 is 28; to 28 we add 3 and 4 (corresponding to the prime divisors 2 and 3) bringing us up to 35; for 4 and 6 we continue the recursion, with 4 bringing us up to 45 and 6 brings up to 64.
%p A191150 a:= proc(n) option remember; uses numtheory;
%p A191150       sigma(n)+add(a(d), d=divisors(n) minus {1,n})
%p A191150     end:
%p A191150 seq(a(n), n=1..100);  # _Alois P. Heinz_, Aug 01 2023
%t A191150 hyperSigma[1] := 1; hyperSigma[n_] := hyperSigma[n] = Module[{d=Divisors[n]}, Total[d] + Total[hyperSigma /@ Rest[Most[d]]]]; Table[hyperSigma[n], {n, 100}] (* From _T. D. Noe_ with a slight modification *)
%o A191150 (PARI) A191150(n) = (sigma(n)+sumdiv(n,d,if((d>1)&&(d<n), A191150(d), 0))); \\ (after the Maple-program) - _Antti Karttunen_, Nov 22 2024
%Y A191150 Cf. A000040, A000203, A008864, A074206, A378210 (Dirichlet inverse).
%Y A191150 Cf. also A191161 (variant 2 of hypersigma).
%K A191150 nonn,easy
%O A191150 1,2
%A A191150 _Alonso del Arte_, May 26 2011
%E A191150 Example corrected by _Paolo P. Lava_, Jul 13 2011