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Partial sums of A347870; Number of terms of A347877 (numbers k with an odd arithmetic derivative of sigma(k)) in range 1..n.
4

%I #6 Feb 14 2022 11:21:30

%S 0,1,1,2,3,3,3,3,4,5,5,5,6,6,6,7,8,8,8,9,9,9,9,9,10,11,11,11,12,12,12,

%T 13,13,14,14,14,15,15,15,16,17,17,17,17,18,18,18,18,18,18,18,19,20,20,

%U 20,20,20,21,21,21,22,22,22,23,23,23,23,24,24,24,24,25,26,27,27,27,27,27,27,28,28,29,29,29,29

%N Partial sums of A347870; Number of terms of A347877 (numbers k with an odd arithmetic derivative of sigma(k)) in range 1..n.

%C Density of terms of A347877 in N seem to be steadily decreasing, e.g. for a(8)/8 = 3/8 = 0.375, a(1024)/1024 = 249/1024 = 0.243..., and a(2^20)/2^20 = 117216/1048576 = 0.111786...

%F a(1) = 0; for n > 1, a(n) = A347870(n) + a(n-1).

%F For all n >= 1, a(A347877(n)) = n.

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A342925(n) = A003415(sigma(n));

%o A347870(n) = (A342925(n)%2);

%o A349909(n) = if(1==n,A347870(n),A347870(n)+A349909(n-1));

%Y Cf. A000203, A003415, A342925, A347870, A347877, A349909.

%K nonn

%O 1,4

%A _Antti Karttunen_, Feb 13 2022