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a(n) = n*sigma(n) - Sum_{(d<n)|n} d*sigma(d).
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%I #27 Nov 10 2024 05:45:54

%S 1,5,11,21,29,53,55,85,104,143,131,217,181,273,317,341,305,494,379,

%T 595,603,653,551,865,744,903,950,1141,869,1499,991,1365,1439,1523,

%U 1593,2002,1405,1893,1989,2395,1721,2877,1891,2737,2990,2753,2255,3441,2736,3658

%N a(n) = n*sigma(n) - Sum_{(d<n)|n} d*sigma(d).

%C If d are divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes sigma(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = n*sigma(n) - Sum_{(d<n)|n} d*sigma(d).

%C Sequence of numbers n such that a(n) = a(k) has solution for distinct numbers n and k: 314, 329, 411, 427, ...

%H Jaroslav Krizek, <a href="/A245773/b245773.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = 2*A064987(n) - A001001(n) = 2*n*sigma(n) - Sum_{d|n} d*sigma(d).

%F a(n) = A064987(n) - A245484(n).

%F a(n) > 1 for all n >= 1.

%e For n=6 with divisors [1,2,3,6] we have: a(6) = 6*sigma(6)-(3*sigma(3)+2*sigma(2)+1*sigma(1)) = 6*12-(3*4+2*3+1*1) = 53.

%p a:= proc(n) n * sigma(n) - add(d*sigma(d), d = divisors(n) minus {n}) end proc:

%p seq(a(n),n=1..100); # _Robert Israel_, Aug 17 2014

%t a245773[n_Integer] := n*DivisorSigma[1, n] - Total[#*DivisorSigma[1, #] & /@ Most[Divisors[n]]]; a245773 /@ Range[50] (* _Michael De Vlieger_, Aug 17 2014 *)

%o (Magma) [(2*n*SumOfDivisors(n)-(&+[d*SumOfDivisors(d): d in Divisors(n)])): n in [1..1000]];

%Y Cf. A000203, A001001, A064987, A245484.

%K nonn,changed

%O 1,2

%A _Jaroslav Krizek_, Aug 16 2014