A353155 Logarithmic transform of odd primes.
3, -4, 16, -94, 740, -7322, 87096, -1209242, 19190176, -342623408, 6797028096, -148325493672, 3531032617412, -91064679012376, 2529198638215228, -75262590212948118, 2388933783463085676, -80567150574145456164, 2876970976034496438802, -108441134639989639371264
Offset: 1
Keywords
Programs
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Maple
a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j* binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->ithprime(i+1)) end: seq(a(n), n=1..25); # Alois P. Heinz, Apr 27 2022 -
Mathematica
nmax = 20; CoefficientList[Series[Log[1 + Sum[Prime[k + 1] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! // Rest a[n_] := a[n] = Prime[n + 1] - (1/n) Sum[Binomial[n, k] Prime[n - k + 1] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
Formula
E.g.f.: log( 1 + Sum_{k>=1} prime(k+1) * x^k / k! ).
a(n) = prime(n+1) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * prime(n-k+1) * k * a(k).