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A363734 a(n) = Sum_{k=0..n} n^divides(k, n), where divides(k, n) = 1 if k divides n, otherwise 0.

%I A363734 #26 Sep 20 2024 14:41:50
%S A363734 0,2,5,8,14,14,27,20,37,34,47,32,79,38,67,72,92,50,121,56,135,102,107,
%T A363734 68,209,98,127,132,191,86,263,92,219,162,167,172,352,110,187,192,353,
%U A363734 122,371,128,303,310,227,140,519,194,345,252,359,158,479,272,497
%N A363734 a(n) = Sum_{k=0..n} n^divides(k, n), where divides(k, n) = 1 if k divides n, otherwise 0.
%H A363734 Peter Luschny, <a href="/A363734/b363734.txt">Table of n, a(n) for n = 0..10000</a>
%F A363734 a(n) = (n - 1) * tau(n) + n + 1 for n >= 1, where tau = A000005.
%F A363734 a(n) + A363735(n) = (n + 1)^2.
%F A363734 A363735(n) - a(n) = A363421(n).
%t A363734 A363734[n_]:=If[n==0,0,n+1+(n-1)DivisorSigma[0,n]];Array[A363734,100,0] (* _Paolo Xausa_, Aug 06 2023 *)
%o A363734 (SageMath)
%o A363734 print([sum(n^k.divides(n) for k in srange(n+1)) for n in srange(57)])
%o A363734 (Python)
%o A363734 from sympy import divisor_count
%o A363734 def A363734(n): return (n-1)*divisor_count(n)+n+1 if n else 0 # _Chai Wah Wu_, Jun 28 2023
%Y A363734 Cf. A113704, A000005, A363735, A363421.
%K A363734 nonn
%O A363734 0,2
%A A363734 _Peter Luschny_, Jun 27 2023