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a(n) = Number of terms of A206074 in range 0 .. n.
9

%I #17 Dec 20 2021 09:47:06

%S 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 12,12,12,12,12,12,13,13,13,13,14,14,15,15,15,15,16,16,16,16,16,16,17,

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

%H Antti Karttunen, <a href="/A255574/b255574.txt">Table of n, a(n) for n = 0..65537</a>

%F a(0) = 0; for n >= 1, a(n) = A257000(n) + a(n-1).

%F Other identities and observations.

%F For all n >= 0:

%F a(n) = n - A255573(n).

%F For all n >= 1:

%F a(A206074(n)) = n. [This sequence works as a left inverse for injection A206074.]

%F a(n) >= A000720(n). [Because primes is a subsequence of A206074.]

%F a(n) >= A091226(n). [Because A014580 is a subsequence of A206074.]

%t binPol[n_, x_] := With[{bb = IntegerDigits[n, 2]}, bb.x^Range[Length[bb]-1, 0, -1]];

%t b[n_] := If[IrreduciblePolynomialQ[binPol[n, x]], 1, 0];

%t b /@ Range[0, 128] // Accumulate (* _Jean-François Alcover_, Dec 20 2021 *)

%o (PARI)

%o isA206074(n) = polisirreducible(Pol(binary(n)));

%o 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_write_bfile(65537);

%o (Scheme) (definec (A255574 n) (if (zero? n) n (+ (A257000 n) (A255574 (- n 1)))))

%Y Partial sums of A257000.

%Y Cf. A000720, A014580, A091226, A206074, A255572, A255573.

%K nonn

%O 0,4

%A _Antti Karttunen_, May 14 2015