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%I #9 May 08 2022 08:45:00
%S 1,2,1,4,1,0,-1,11,-3,-7,4,-10,5,-15,-4,151,-9,-50,-3,-63,19,-176,6,
%T 591,-27,-637,13,-999,50,-1957,-49,27250,-162,-7887,83,-12821,468,
%U -27226,-40,127341,-1215,-99166,-526,-174140,2640,-362870,1673,1419061,-4516,-1344620
%N Product_{n>=1} (1 + a(n)*x^n) = 1 + x + Sum_{n>=2} prime(n-1)*x^n.
%t nn = 50; f[x_] := Product[(1 + a[n] x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - x - Sum[Prime[k - 1] x^k, {k, 2, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
%Y Cf. A008578, A147557, A353605.
%K sign
%O 1,2
%A _Ilya Gutkovskiy_, May 07 2022