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A347136 a(n) = Sum_{d|n} d * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes.

%I A347136 #20 Oct 18 2021 14:11:26
%S A347136 1,5,8,19,12,40,18,65,49,60,24,152,30,90,96,211,36,245,42,228,144,120,
%T A347136 52,520,109,150,272,342,60,480,68,665,192,180,216,931,78,210,240,780,
%U A347136 84,720,90,456,588,260,100,1688,247,545,288,570,112,1360,288,1170,336,300,120,1824,128,340,882,2059,360,960,138
%N A347136 a(n) = Sum_{d|n} d * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes.
%C A347136 Dirichlet convolution of the identity function (A000027) with the prime shifted identity (A003961). Multiplicative because both A000027 and A003961 are.
%C A347136 Dirichlet convolution of Euler phi (A000010) with the prime shifted sigma (A003973).
%C A347136 Dirichlet convolution of sigma (A000203) with the prime shifted phi (A003972).
%C A347136 Inverse Möbius transform of A347137.
%H A347136 Antti Karttunen, <a href="/A347136/b347136.txt">Table of n, a(n) for n = 1..10000</a>
%H A347136 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A347136 a(n) = Sum_{d|n} d * A003961(n/d).
%F A347136 a(n) = Sum_{d|n} A000010(n/d) * A003973(d).
%F A347136 a(n) = Sum_{d|n} A000203(n/d) * A003972(d).
%F A347136 a(n) = Sum_{d|n} A347137(d).
%F A347136 For all primes p, a(p) = p + A003961(p).
%F A347136 a(n) = A347121(n) + 2*n.
%F A347136 Multiplicative with a(p^e) = (A151800(p)^(e+1) - p^(e+1))/(A151800(p)-p). - _Amiram Eldar_, Aug 24 2021
%t A347136 f[p_, e_] := ((np = NextPrime[p])^(e + 1) - p^(e + 1))/(np - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 24 2021 *)
%o A347136 (PARI)
%o A347136 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A347136 A347136(n) = sumdiv(n,d,d*A003961(n/d));
%Y A347136 Cf. A003961, A003972, A003973, A151800, A347121, A347137 (Möbius transform).
%Y A347136 Cf. also A038040, A336845, A336475, A347236.
%K A347136 nonn,mult
%O A347136 1,2
%A A347136 _Antti Karttunen_, Aug 24 2021