A349909 Partial sums of A347870; Number of terms of A347877 (numbers k with an odd arithmetic derivative of sigma(k)) in range 1..n.

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

Offset: 1

Author

Antti Karttunen, Feb 13 2022

Keywords

• nonn

Comments

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...

Crossrefs

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

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));
A347870(n) = (A342925(n)%2);
A349909(n) = if(1==n,A347870(n),A347870(n)+A349909(n-1));

Formula

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

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