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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.
5
1, 5, 8, 19, 12, 40, 18, 65, 49, 60, 24, 152, 30, 90, 96, 211, 36, 245, 42, 228, 144, 120, 52, 520, 109, 150, 272, 342, 60, 480, 68, 665, 192, 180, 216, 931, 78, 210, 240, 780, 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
OFFSET
1,2
COMMENTS
Dirichlet convolution of the identity function (A000027) with the prime shifted identity (A003961). Multiplicative because both A000027 and A003961 are.
Dirichlet convolution of Euler phi (A000010) with the prime shifted sigma (A003973).
Dirichlet convolution of sigma (A000203) with the prime shifted phi (A003972).
Inverse Möbius transform of A347137.
FORMULA
a(n) = Sum_{d|n} d * A003961(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A003973(d).
a(n) = Sum_{d|n} A000203(n/d) * A003972(d).
a(n) = Sum_{d|n} A347137(d).
For all primes p, a(p) = p + A003961(p).
a(n) = A347121(n) + 2*n.
Multiplicative with a(p^e) = (A151800(p)^(e+1) - p^(e+1))/(A151800(p)-p). - Amiram Eldar, Aug 24 2021
MATHEMATICA
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 *)
PROG
(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347136(n) = sumdiv(n, d, d*A003961(n/d));
CROSSREFS
Cf. A003961, A003972, A003973, A151800, A347121, A347137 (Möbius transform).
Sequence in context: A302393 A342804 A226902 * A326826 A026595 A037233
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Aug 24 2021
STATUS
approved