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A255574 a(n) = Number of terms of A206074 in range 0 .. n.

%I A255574 #17 Dec 20 2021 09:47:06
%S A255574 0,0,1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,10,10,10,10,11,11,
%T A255574 12,12,12,12,12,12,13,13,13,13,14,14,15,15,15,15,16,16,16,16,16,16,17,
%U A255574 17,18,18,18,18,19,19,20,20,20,20,20,20,21,21,22,22,23,23,24,24,24,24,25,25,26,26,27,27
%N A255574 a(n) = Number of terms of A206074 in range 0 .. n.
%H A255574 Antti Karttunen, <a href="/A255574/b255574.txt">Table of n, a(n) for n = 0..65537</a>
%F A255574 a(0) = 0; for n >= 1, a(n) = A257000(n) + a(n-1).
%F A255574 Other identities and observations.
%F A255574 For all n >= 0:
%F A255574 a(n) = n - A255573(n).
%F A255574 For all n >= 1:
%F A255574 a(A206074(n)) = n. [This sequence works as a left inverse for injection A206074.]
%F A255574 a(n) >= A000720(n). [Because primes is a subsequence of A206074.]
%F A255574 a(n) >= A091226(n). [Because A014580 is a subsequence of A206074.]
%t A255574 binPol[n_, x_] := With[{bb = IntegerDigits[n, 2]}, bb.x^Range[Length[bb]-1, 0, -1]];
%t A255574 b[n_] := If[IrreduciblePolynomialQ[binPol[n, x]], 1, 0];
%t A255574 b /@ Range[0, 128] // Accumulate (* _Jean-François Alcover_, Dec 20 2021 *)
%o A255574 (PARI)
%o A255574 isA206074(n) = polisirreducible(Pol(binary(n)));
%o A255574 A255574_write_bfile(up_to_n) = { my(n,a_n=0); for(n=0, up_to_n, if(isA206074(n),a_n++); write("b255574.txt", n, " ", a_n)); };
%o A255574 A255574_write_bfile(65537);
%o A255574 (Scheme) (definec (A255574 n) (if (zero? n) n (+ (A257000 n) (A255574 (- n 1)))))
%Y A255574 Partial sums of A257000.
%Y A255574 Cf. A000720, A014580, A091226, A206074, A255572, A255573.
%K A255574 nonn
%O A255574 0,4
%A A255574 _Antti Karttunen_, May 14 2015