login
a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).
4

%I #11 Oct 29 2021 07:18:29

%S 0,0,0,2,0,0,0,10,3,0,0,6,0,0,0,38,0,6,0,10,0,0,0,30,5,0,21,14,0,0,0,

%T 130,0,0,0,36,0,0,0,50,0,0,0,22,15,0,0,114,7,10,0,26,0,42,0,70,0,0,0,

%U 30,0,0,21,422,0,0,0,34,0,0,0,144,0,0,15,38,0,0,0,190,111,0,0,42,0,0,0,110,0,30,0,46

%N a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

%C Möbius transform of A348029(n), which is A003959(n) - sigma(n).

%H Antti Karttunen, <a href="/A348030/b348030.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = A003968(n) - n.

%F a(n) = Sum_{d|n} A008683(n/d) * A348029(d).

%t f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* _Amiram Eldar_, Oct 20 2021 *)

%o (PARI)

%o A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }

%o A348030(n) = (A003968(n)-n);

%Y Cf. A003959, A003968, A005117 (positions of zeros), A008683, A348029, A348036.

%K nonn

%O 1,4

%A _Antti Karttunen_, Oct 19 2021