%I #31 Nov 16 2021 08:24:18

%S 0,0,0,1,0,1,0,7,1,1,0,8,0,1,1,33,0,9,0,10,1,1,0,40,1,1,10,12,0,11,0,

%T 131,1,1,1,48,0,1,1,54,0,13,0,16,12,1,0,164,1,13,1,18,0,57,1,68,1,1,0,

%U 64,0,1,14,473,1,17,0,22,1,15,0,204,0,1,14,24,1,19,0,230,67,1,0,80,1,1,1,96,0,75,1,28

%N a(n) = A003959(n) - A129283(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

%C There are no negative terms. We prove this by induction over the prime factorization of n, showing that A348507(n) >= A003415(n) for all values of n >= 1. At n=1, both sequences have value 0, and at the primes both sequences obtain the value 1, so the base cases hold. We know that A348507(n)-(n/p) = (p+1)*A348507(n/p) for all prime factors p of n (see comment in A348507). With the arithmetic derivative we obtain respectively that A003415(n) = A003415(p*(n/p)) = A003415(p)*(n/p) + p*A003415(n/p) = (n/p) + p*A003415(n/p), for any prime factor p of n. Now A348507(p*(n/p)) >= A003415(p*(n/p)) iff A348507(p*(n/p)) - (n/p) >= A003415(p*(n/p)) - (n/p), that is, iff (p+1)*A348507(n/p) >= p*A003415(n/p), which indeed follows by the induction hypothesis, which assumes that A348507(x) >= A003415(x) for all proper divisors x of n.

%H Antti Karttunen, <a href="/A348970/b348970.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A348970/a348970.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = A003959(n) - A129283(n) = A003959(n) - (n+A003415(n)).

%F a(n) = A348029(n) - A211991(n).

%F a(n) = A348507(n) - A003415(n).

%F For all n >= 1, a(A001358(n)) = 1.

%t d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n - d[n]; Array[a, 100] (* _Amiram Eldar_, Nov 05 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 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A348970(n) = (A003959(n) - (n+A003415(n)));

%Y Cf. A003415, A003959, A129283, A211991, A348029, A348507, A348971, A348972, A348973, A348974.

%Y Cf. A008578 (positions of zeros), A001358 (positions of ones).

%K nonn

%O 1,8

%A _Antti Karttunen_, Nov 05 2021