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a(n) = Sum_{d|n} phi(d) * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes, and phi is Euler totient function.
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%I #19 Sep 16 2023 05:25:40

%S 1,4,7,14,11,28,17,46,41,44,23,98,29,68,77,146,35,164,41,154,119,92,

%T 51,322,97,116,223,238,59,308,67,454,161,140,187,574,77,164,203,506,

%U 83,476,89,322,451,204,99,1022,229,388,245,406,111,892,253,782,287,236,119,1078,127,268,697,1394,319,644,137,490

%N a(n) = Sum_{d|n} phi(d) * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes, and phi is Euler totient function.

%C Dirichlet convolution of Euler phi (A000010) with the prime shift function (A003961). Multiplicative because both A000010 and A003961 are.

%C Dirichlet convolution of the identity function (A000027) with the prime shifted phi (A003972).

%C Möbius transform of A347136.

%H Antti Karttunen, <a href="/A347137/b347137.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%F a(n) = Sum_{d|n} A000010(n/d) * A003961(d).

%F a(n) = Sum_{d|n} d * A003972(n/d).

%F a(n) = Sum_{d|n} A008683(n/d) * A347136(d).

%F a(n) = A347122(n) + 2*A000010(n).

%F a(A000040(n)) = A001043(n) - 1.

%F Multiplicative with a(p^e) = q(p)^e + (p-1)*(q(p)^e - p^e)/(q(p) - p), where q(p) = nextprime(p) = A151800(p). - _Amiram Eldar_, Sep 16 2023

%t f[p_, e_] := (q = NextPrime[p])^e + (p - 1)*(q^e - p^e)/(q - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 16 2023 *)

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A347137(n) = sumdiv(n,d,eulerphi(n/d)*A003961(d));

%Y Cf. A000010, A000027, A000040, A001043, A003961, A003972, A008683, A151800, A347122, A347136 (inverse Möbius transform).

%Y Cf. also A018804, A347237.

%K nonn,easy,mult

%O 1,2

%A _Antti Karttunen_, Aug 24 2021