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A340976 Sum_{1 < k < n} sigma(n) mod k, where sigma = A000203.

%I A340976 #25 Feb 03 2021 23:27:00
%S A340976 0,0,0,2,2,2,7,8,18,11,16,27,30,30,40,47,46,75,60,72,101,93,84,109,
%T A340976 146,148,167,142,137,180,166,197,254,282,283,301,247,333,367,347,283,
%U A340976 389,327,367,475,501,373,591,517,562,621,597,491,615,699,637,810,839,585,783,671,964,1024
%N A340976 Sum_{1 < k < n} sigma(n) mod k, where sigma = A000203.
%C A340976 Motivated by A340180 and several other sequences that use the sum over a subset of the indices.
%C A340976 Is there an efficient formula for a(n)? That might answer the following questions:
%C A340976 1) Is a(63) = a(2^6-1) = 1024 = 2^10 just a coincidence?
%C A340976 2) Are there are further terms of the form 2^k, i.e., a(n) in A000079? What can be said about these n?
%C A340976 3) Are there other fixed points a(n) = n as for n = 7, 8?
%C A340976 4) What is the frequency of odd vs. even terms? a(n) is odd for consecutive indices 21..22, 35..49, 51..56, 58..61, 64..65, 68..69, 73..79, ...: Are there patterns or simple subsequence(s) of such runs of length 2 or larger?
%F A340976 a(n) = (n-1)*sigma(n) - A024916(sigma(n)) + Sum_{k=n..sigma(n)} k*floor(sigma(n)/k). - _Daniel Suteu_, Feb 02 2021
%t A340976 Table[Sum[Mod[DivisorSigma[1,n],k],{k,2,n-1}],{n,1,138}] (* _Metin Sariyar_, Feb 02 2021 *)
%o A340976 (PARI) apply( {A340976(n,s=sigma(n))=sum(k=1,n-1,s%k)}, [1..66]) \\ _M. F. Hasler_, Feb 01 2021
%o A340976 (PARI)
%o A340976 T(n) = n*(n+1)/2;
%o A340976 S(n) = my(s=sqrtint(n)); sum(k=1, s, T(n\k) + k*(n\k)) - s*T(s); \\ A024916
%o A340976 g(a,b) = my(s=0); while(a <= b, my(t=b\a); my(u=b\t); s += t*(T(u) - T(a-1)); a = u+1); s;
%o A340976 a(n) = (n-1)*sigma(n) - S(sigma(n)) + g(n, sigma(n)); \\ _Daniel Suteu_, Feb 02 2021
%Y A340976 Cf. A000203, A340179, A340180.
%K A340976 nonn
%O A340976 1,4
%A A340976 _M. F. Hasler_, Feb 01 2021