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

0, 0, 0, 2, 2, 2, 7, 8, 18, 11, 16, 27, 30, 30, 40, 47, 46, 75, 60, 72, 101, 93, 84, 109, 146, 148, 167, 142, 137, 180, 166, 197, 254, 282, 283, 301, 247, 333, 367, 347, 283, 389, 327, 367, 475, 501, 373, 591, 517, 562, 621, 597, 491, 615, 699, 637, 810, 839, 585, 783, 671, 964, 1024

Offset: 1

Author

M. F. Hasler, Feb 01 2021

Keywords

• nonn

Comments

Motivated by A340180 and several other sequences that use the sum over a subset of the indices.

Is there an efficient formula for a(n)? That might answer the following questions:

1) Is a(63) = a(2^6-1) = 1024 = 2^10 just a coincidence?

2) Are there are further terms of the form 2^k, i.e., a(n) in A000079? What can be said about these n?

3) Are there other fixed points a(n) = n as for n = 7, 8?

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?

Crossrefs

Cf. A000203, A340179, A340180.

Programs

Mathematica

Table[Sum[Mod[DivisorSigma[1,n],k],{k,2,n-1}],{n,1,138}] (* Metin Sariyar, Feb 02 2021 *)

PARI

apply( {A340976(n,s=sigma(n))=sum(k=1,n-1,s%k)}, [1..66]) \\ M. F. Hasler, Feb 01 2021

PARI

T(n) = n*(n+1)/2;
S(n) = my(s=sqrtint(n)); sum(k=1, s, T(n\k) + k*(n\k)) - s*T(s); \\ A024916
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;
a(n) = (n-1)*sigma(n) - S(sigma(n)) + g(n, sigma(n)); \\ Daniel Suteu, Feb 02 2021

Formula

a(n) = (n-1)*sigma(n) - A024916(sigma(n)) + Sum_{k=n..sigma(n)} k*floor(sigma(n)/k). - Daniel Suteu, Feb 02 2021