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A347137
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.
4
1, 4, 7, 14, 11, 28, 17, 46, 41, 44, 23, 98, 29, 68, 77, 146, 35, 164, 41, 154, 119, 92, 51, 322, 97, 116, 223, 238, 59, 308, 67, 454, 161, 140, 187, 574, 77, 164, 203, 506, 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
OFFSET
1,2
COMMENTS
Dirichlet convolution of Euler phi (A000010) with the prime shift function (A003961). Multiplicative because both A000010 and A003961 are.
Dirichlet convolution of the identity function (A000027) with the prime shifted phi (A003972).
Möbius transform of A347136.
FORMULA
a(n) = Sum_{d|n} A000010(n/d) * A003961(d).
a(n) = Sum_{d|n} d * A003972(n/d).
a(n) = Sum_{d|n} A008683(n/d) * A347136(d).
a(n) = A347122(n) + 2*A000010(n).
a(A000040(n)) = A001043(n) - 1.
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
MATHEMATICA
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 *)
PROG
(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347137(n) = sumdiv(n, d, eulerphi(n/d)*A003961(d));
CROSSREFS
Cf. A000010, A000027, A000040, A001043, A003961, A003972, A008683, A151800, A347122, A347136 (inverse Möbius transform).
Cf. also A018804, A347237.
Sequence in context: A074862 A101064 A243707 * A347413 A310825 A062380
KEYWORD
nonn,easy,mult
AUTHOR
Antti Karttunen, Aug 24 2021
STATUS
approved