A342925 - OEIS

%I #17 Feb 25 2022 16:58:35

%S 0,1,4,1,5,16,12,8,1,21,16,32,9,44,44,1,21,16,24,41,80,60,44,92,1,41,

%T 68,92,31,156,80,51,112,81,112,20,21,92,92,123,41,272,48,124,71,156,

%U 112,128,22,34,156,77,81,244,156,244,176,123,92,332,33,272,164,1,124,384,72,165,272,384,156,119,39,101,128,188

%N a(n) = A003415(sigma(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

%H Antti Karttunen, <a href="/A342925/b342925.txt">Table of n, a(n) for n = 1..16384</a>

%F a(A023194(n)) = 1.

%F If gcd(m,n) = 1, a(m*n) = sigma(m)*A003415(sigma(n)) + sigma(n)*A003415(sigma(m)) = sigma(m)*a(n) + sigma(n)*a(m).

%F a(n) = (A351568(n)*A351571(n)) + (A351569(n)*A351570(n)). - _Antti Karttunen_, Feb 23 2022

%t Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ DivisorSigma[1, #] &, 76] (* _Michael De Vlieger_, Apr 08 2021 *)

%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));

%Y Cf. A023194 (positions of ones, which is a subsequence of prime powers, A000961).

%Y Cf. A342922, A342923, A342924, A342926, A351568, A351569, A351570, A351571.

%Y Cf. A342021 (fixed points), A343216 [positions k where a(k) < k], A343217 [a(k) >= k], A343218 [a(k) > k].

%Y Cf. A347870 (parity of terms), A347872, A347873, A347877 (positions of odd terms), A347878 (of even terms), A343218, A343220, A344024.

%K nonn

%O 1,3

%A _Antti Karttunen_, Apr 07 2021