%I #44 Nov 11 2021 19:30:45

%S 0,1,1,5,1,6,1,19,7,8,1,24,1,10,9,65,1,30,1,34,11,14,1,84,11,16,37,44,

%T 1,42,1,211,15,20,13,108,1,22,17,122,1,54,1,64,51,26,1,276,15,58,21,

%U 74,1,138,17,160,23,32,1,156,1,34,65,665,19,78,1,94,27,74,1,360,1,40,69,104,19,90,1,406,175,44,1,204

%N a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

%C a(p*(n/p)) - (n/p) = (p+1)*a(n/p) holds for all prime divisors p of n, which can be seen by expanding the left hand side as (A003959(p*(n/p)) - (p*(n/p))) - (n/p) = (p+1)*A003959(n/p)-((p+1)*(n/p)) = (p+1)*(A003959(n/p)-(n/p)) = (p+1)*a(n/p). This implies that a(n) >= A003415(n) for all n. (See also comments in A348970). - _Antti Karttunen_, Nov 06 2021

%H Antti Karttunen, <a href="/A348507/b348507.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A348507/a348507.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = A003959(n) - n.

%F a(n) = A348508(n) + n.

%F a(n) = A001065(n) + A348029(n).

%F From _Antti Karttunen_, Nov 06 2021: (Start)

%F a(n) = Sum_{d|n} A348971(d).

%F a(n) = A003415(n) + A348970(n).

%F For all n >= 1, A322582(n) <= A003415(n) <= a(n).

%F For n > 1, a(n) = a(A032742(n))*(1+A020639(n)) + A032742(n). [See the comments above, and compare this with _Reinhard Zumkeller_'s May 09 2011 recursive formula for A003415]

%F (End)

%t f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* _Amiram Eldar_, Oct 30 2021 *)

%o (PARI)

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

%o A348507(n) = (A003959(n) - n);

%o (PARI)

%o A020639(n) = if(1==n,n,(factor(n)[1, 1]));

%o A348507(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= (1+spf)); (s); }; \\ (Compare this with similar programs given in A003415 and in A322582) - _Antti Karttunen_, Nov 06 2021

%Y Cf. A001065, A003415, A003959, A020639, A032742, A348029, A348508, A348929 [= gcd(n,a(n))], A348950, A348970.

%Y Cf. A348971 (Möbius transform) and A349139, A349140, A349141, A349142, A349143 (other Dirichlet convolutions).

%Y Cf. also A168065 (the arithmetic mean of this and A322582), A168066.

%K nonn

%O 1,4

%A _Antti Karttunen_, Oct 30 2021