This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
f:= proc(n) local d; add((-1)^(n/d+1)*d*ithprime(d), d = numtheory:-divisors(n)); end proc:map(f, [$1..100]); # Robert Israel, Aug 01 2023
Table[Sum[(-1)^(n/d + 1) d Prime[d], {d, Divisors[n]}], {n, 50}]
nmax = 50; Rest[CoefficientList[Series[Sum[k Prime[k] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 50; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^Prime[k], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
a(n) = sumdiv(n, d, (-1)^(n/d+1)*d*prime(d)); \\ Michel Marcus, Aug 25 2018
Table[Sum[EulerPhi[d] Prime[d], {d, Divisors[n]}], {n, 1, 50}]
Table[Sum[Prime[n/GCD[n, k]], {k, 1, n}], {n, 1, 50}]
a(n) = sumdiv(n, d, prime(d)*eulerphi(d)); \\ Michel Marcus, Mar 27 2020
a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 1 * 2 + 2 * 3 + 3 * 5 = 23.
nmax = 65; CoefficientList[Series[Sum[k Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]