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A348983 a(n) = Sum_{d|n} A038040(d) * A322582(n/d), where A038040(n) = n*d(n), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

%I A348983 #13 Nov 15 2021 03:40:15
%S A348983 0,1,1,7,1,14,1,31,11,20,1,80,1,26,23,111,1,109,1,122,29,38,1,328,19,
%T A348983 44,76,164,1,250,1,351,41,56,35,565,1,62,47,514,1,334,1,248,208,74,1,
%U A348983 1128,27,245,59,290,1,650,47,700,65,92,1,1336,1,98,274,1023,53,502,1,374,77,490,1,2213,1,116,302,416,53
%N A348983 a(n) = Sum_{d|n} A038040(d) * A322582(n/d), where A038040(n) = n*d(n), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).
%C A348983 Dirichlet convolution of A322582 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
%C A348983 Dirichlet convolution of the identity function (A000027) with A348980.
%C A348983 Dirichlet convolution of sigma (A000203) with A348981.
%H A348983 Antti Karttunen, <a href="/A348983/b348983.txt">Table of n, a(n) for n = 1..16384</a>
%F A348983 a(n) = Sum_{d|n} A038040(n/d) * A322582(d).
%F A348983 a(n) = Sum_{d|n} d * A348980(n/d).
%F A348983 a(n) = Sum_{d|n} A000203(d) * A348981(n/d).
%F A348983 For all n >= 1, a(n) <= A349123(n) <= A349143(n).
%t A348983 f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)
%o A348983 (PARI)
%o A348983 A038040(n) = (n*numdiv(n));
%o A348983 A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
%o A348983 A322582(n) = (n-A003958(n));
%o A348983 A348983(n) = sumdiv(n,d,A038040(n/d)*A322582(d));
%Y A348983 Cf. A000005, A000027, A000203, A003958, A038040, A322582, A348980, A348981, A348982.
%Y A348983 Cf. also A349123, A349143.
%K A348983 nonn
%O A348983 1,4
%A A348983 _Antti Karttunen_, Nov 08 2021