A348945 - OEIS

%I #13 Feb 11 2022 11:23:43

%S 0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,12,0,0,6,0,0,0,0,75,0,

%T 0,0,6,0,0,0,18,0,0,0,0,0,0,0,72,0,0,0,0,0,18,0,24,0,0,0,0,0,0,0,270,

%U 0,0,0,0,0,0,0,66,0,0,0,0,0,0,0,108,48,0,0,0,0,0,0,36,0,0,0,0,0,0,0,300,0,0,0,10

%N a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

%H Antti Karttunen, <a href="/A348945/b348945.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A348944(n) - A000203(n) = ((1/2) * (A003959(n)+A034448(n))) - A000203(n).

%F a(n) = (1/2) * (A348029(n)-A048146(n)).

%t f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 0; a[n_] := (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2 - Times @@ f1 @@@ f; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)

%o (PARI)

%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };

%o A348944(n) = ((1/2)*(A003959(n)+A034448(n)));

%o A348945(n) = (A348944(n)-sigma(n));

%Y Cf. A000203, A003959, A034448, A048146, A348029, A348944, A348946.

%K nonn

%O 1,8

%A _Antti Karttunen_, Nov 05 2021