cp's OEIS Frontend

A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

%I A349140 #14 Nov 19 2021 14:05:28
%S A349140 0,1,1,7,1,11,1,33,10,15,1,61,1,19,17,131,1,77,1,89,21,27,1,263,16,31,
%T A349140 67,117,1,145,1,473,29,39,25,379,1,43,33,395,1,189,1,173,137,51,1,997,
%U A349140 22,155,41,201,1,443,33,527,45,63,1,743,1,67,177,1611,37,277,1,257,53,265,1,1541,1,79,187,285,37,321
%N A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).
%C A349140 Dirichlet convolution of A348507 with the identity function, A000027.
%C A349140 Dirichlet convolution of sigma with A348971.
%H A349140 Antti Karttunen, <a href="/A349140/b349140.txt">Table of n, a(n) for n = 1..16384</a>
%F A349140 a(n) = Sum_{d|n} d * A348507(n/d).
%F A349140 a(n) = Sum_{d|n} A000203(d) * A348971(n/d).
%F A349140 a(n) = Sum_{d|n} A349141(d).
%F A349140 For all n >= 1, a(n) >= A347130(n) >= A348980(n).
%F A349140 a(n) = A349170(n) - A038040(n). - _Antti Karttunen_, Nov 15 2021
%t A349140 f[p_, e_] := (p + 1)^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, #*s[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)
%o A349140 (PARI)
%o A349140 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A349140 A348507(n) = (A003959(n) - n);
%o A349140 A349140(n) = sumdiv(n,d,d*A348507(n/d));
%Y A349140 Cf. A000027, A000203, A003959, A038040, A348507, A348971, A349141 (Möbius transform), A349142, A349143, A349170.
%Y A349140 Cf. also A347130, A348980.
%K A349140 nonn
%O A349140 1,4
%A A349140 _Antti Karttunen_, Nov 08 2021