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.
a(4)=36 because the divisors of 4 are 1,2,4 and 1*p(1) + 2*p(2) + 4*p(4) = 1*2 + 2*3 + 4*7 = 36.
with(numtheory): a:=proc(n) local div: div:=divisors(n): sum(div[j]*ithprime(div[j]),j=1..tau(n)) end: seq(a(n),n=1..55); # Emeric Deutsch, Jan 20 2007
a(n) = sumdiv(n, d, d*prime(d)); \\ Michel Marcus, Jun 24 2018
a(6) = 3 because we have [3a, 3b], [3a, 3c] and [3b, 3c].
nmax = 50; CoefficientList[Series[Product[(1 + x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
end:
a:= proc(n) option remember; `if`(n=0, 1,
-add(b(n-i$2)*a(i$2), i=0..n-1))
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jun 13 2018
nmax = 33; CoefficientList[Series[Product[1/(1 + Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[Sum[Sum[(-1)^k Prime[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-Prime[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]
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
Comments