cp's OEIS Frontend

A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

%I A349142 #13 Nov 15 2021 03:40:19
%S A349142 0,1,1,8,1,13,1,40,11,17,1,80,1,21,19,164,1,99,1,112,23,29,1,364,17,
%T A349142 33,77,144,1,191,1,604,31,41,27,528,1,45,35,524,1,243,1,208,165,53,1,
%U A349142 1424,23,187,43,240,1,597,35,684,47,65,1,1072,1,69,209,2084,39,347,1,304,55,327,1,2244,1,81,221,336,39,399
%N A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).
%C A349142 Dirichlet convolution of A001615 with A348507.
%H A349142 Antti Karttunen, <a href="/A349142/b349142.txt">Table of n, a(n) for n = 1..16384</a>
%F A349142 a(n) = Sum_{d|n} A001615(n/d) * A348507(d).
%F A349142 For all n >= 1, a(n) >= A347132(n) >= A348982(n).
%F A349142 a(n) = A349172(n) - A327251(n). - _Antti Karttunen_, Nov 14 2021
%t A349142 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 A349142 (PARI)
%o A349142 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 A349142 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A349142 A348507(n) = (A003959(n) - n);
%o A349142 A349142(n) = sumdiv(n,d,A001615(d)*A348507(n/d));
%Y A349142 Cf. A001615, A003959, A327251, A348507, A349140, A349141, A349143, A349172.
%Y A349142 Cf. also A347132, A348982.
%K A349142 nonn
%O A349142 1,4
%A A349142 _Antti Karttunen_, Nov 08 2021