A348982 - OEIS

%I #12 Nov 15 2021 03:40:31

%S 0,1,1,6,1,11,1,22,9,15,1,52,1,19,17,66,1,69,1,76,21,27,1,176,15,31,

%T 51,100,1,145,1,178,29,39,25,288,1,43,33,264,1,189,1,148,123,51,1,508,

%U 21,145,41,172,1,339,33,352,45,63,1,632,1,67,159,450,37,277,1,220,53,265,1,924,1,79,175,244,37,321

%N a(n) = Sum_{d|n} psi(n/d) * A322582(d), where psi is Dedekind psi (A001615), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

%C Dirichlet convolution of A001615 with A322582.

%H Antti Karttunen, <a href="/A348982/b348982.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = Sum_{d|n} A001615(n/d) * A322582(d).

%F For all n >= 1, a(n) <= A347132(n) <= A349142(n).

%F a(n) = A327251(n) - A349132(n). - _Antti Karttunen_, Nov 14 2021

%t f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*psi[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)

%o (PARI)

%o A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615

%o A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };

%o A322582(n) = (n-A003958(n));

%o A348982(n) = sumdiv(n,d,A001615(n/d)*A322582(d));

%Y Cf. A001615, A003958, A322582, A327251, A348980, A348981, A348982, A348983, A349132.

%Y Cf. also A347132, A349142.

%K nonn

%O 1,4

%A _Antti Karttunen_, Nov 08 2021