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%I #8 Apr 27 2022 18:40:36
%S 3,-4,16,-94,740,-7322,87096,-1209242,19190176,-342623408,6797028096,
%T -148325493672,3531032617412,-91064679012376,2529198638215228,
%U -75262590212948118,2388933783463085676,-80567150574145456164,2876970976034496438802,-108441134639989639371264
%N Logarithmic transform of odd primes.
%F E.g.f.: log( 1 + Sum_{k>=1} prime(k+1) * x^k / k! ).
%F a(n) = prime(n+1) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * prime(n-k+1) * k * a(k).
%p a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
%p binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->ithprime(i+1))
%p end:
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Apr 27 2022
%t nmax = 20; CoefficientList[Series[Log[1 + Sum[Prime[k + 1] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t 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}]
%Y Cf. A006005, A007447, A065091, A343622, A353079.
%K sign
%O 1,1
%A _Ilya Gutkovskiy_, Apr 27 2022