%I #35 Dec 17 2023 10:26:54
%S 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 37,59,61,112,30,165,32,176,64,75,62,290,38,83,74,252,42,209,44,160,
%U 139,99,48,512,65,166,94,184,54,306,90,316,104,123,60,588,62,131
%N Hypersigma(n): sum of the divisors of n plus the recursive sum of the divisors of the restricted divisors.
%C 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 Thus if n is prime then hypersigma(n) is the same as sigma(n).
%H 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)
%F a(n) = n + 1 <=> n is prime. - _Bill McEachen_, Aug 01 2023
%e 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 a:= proc(n) option remember; uses numtheory;
%p sigma(n)+add(a(d), d=divisors(n) minus {1,n})
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Aug 01 2023
%t 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 *)
%Y Cf. A000040, A000203, A008864.
%K nonn,easy
%O 1,2
%A _Alonso del Arte_, May 26 2011
%E Example corrected by _Paolo P. Lava_, Jul 13 2011