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A091892 Numbers k having only one partition into parts which are a sum of exactly as many distinct powers of 2 as there are 1's in the binary representation of k.

%I A091892 #39 Apr 22 2024 21:44:05
%S A091892 0,1,3,5,7,11,13,15,19,23,27,29,31,39,43,47,51,55,59,61,63,79,87,91,
%T A091892 95,103,107,111,115,119,123,125,127,143,159,175,183,187,191,207,215,
%U A091892 219,223,231,235,239,243,247,251,253,255,287,303,319,335,351,367,375,379,383,399
%N A091892 Numbers k having only one partition into parts which are a sum of exactly as many distinct powers of 2 as there are 1's in the binary representation of k.
%C A091892 All positive terms are odd. - _Alois P. Heinz_, Dec 12 2021
%C A091892 Conjecture: if the second leftmost bit in the binary expansion of k+1 equals 0, then k is a term if and only if A007814(k+1) >= 2^(f(k)-1) + f(k). Otherwise, k is a term if and only if A007814(k+1) >= 2^f(k). Here f(k) = A086784(k+1). - _Mikhail Kurkov_, Oct 03 2022
%H A091892 David A. Corneth, <a href="/A091892/b091892.txt">Table of n, a(n) for n = 1..2053</a> (first 375 terms from Andrew Howroyd, n = 376..764 from Alois P. Heinz)
%F A091892 A091891(a(n)) = 1.
%e A091892 From _David A. Corneth_, Oct 03 2022: (Start)
%e A091892 11 is in the sequence as numbers with 3 bits and are <= 11 are 7, 11. The only partition of 11 into parts of size 7 and 11 are 11.
%e A091892 9 is not in the sequence as numbers with 2 bits, like 9, are 3, 5, 6, 9. 9 can be partitioned as 3+3+3 = 3+6 = 9 into these parts. As these are 3 > 1 partitions, 9 is not here. (End)
%t A091892 etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}] b[n - j], {j, 1, n}]/n]; b];
%t A091892 EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
%t A091892 okQ[k_] := If[k == 0, True, If[EvenQ[k], False, EulerT[Table[DigitCount[j, 2, 1] == DigitCount[k, 2, 1] // Boole, {j, 1, k}]][[k]] == 1]];
%t A091892 Reap[For[k = 0, k <= 1000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Dec 17 2021 *)
%o A091892 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A091892 upto(n)={Set(concat(vector(logint(n,2)+1, k, my(u=vector(n,i,hammingweight(i)==k), v=EulerT(u)); select(i->u[i]&&v[i]==1, [1..n], 1))))} \\ _Andrew Howroyd_, Apr 20 2021
%Y A091892 Cf. A018819, A091889, A091890, A091891, A000120, A000041.
%K A091892 nonn
%O A091892 1,3
%A A091892 _Reinhard Zumkeller_, Feb 10 2004
%E A091892 Terms a(40) and beyond from _Andrew Howroyd_, Apr 20 2021
%E A091892 a(1)=0 inserted by _Alois P. Heinz_, Dec 12 2021