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.

0, 1, 3, 5, 7, 11, 13, 15, 19, 23, 27, 29, 31, 39, 43, 47, 51, 55, 59, 61, 63, 79, 87, 91, 95, 103, 107, 111, 115, 119, 123, 125, 127, 143, 159, 175, 183, 187, 191, 207, 215, 219, 223, 231, 235, 239, 243, 247, 251, 253, 255, 287, 303, 319, 335, 351, 367, 375, 379, 383, 399

Offset: 1

Author

Reinhard Zumkeller, Feb 10 2004

Keywords

• nonn

Comments

All positive terms are odd. - Alois P. Heinz, Dec 12 2021

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

Examples

			From _David A. Corneth_, Oct 03 2022: (Start)
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.
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)
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..2053 (first 375 terms from Andrew Howroyd, n = 376..764 from Alois P. Heinz)

Crossrefs

Cf. A018819, A091889, A091890, A091891, A000120, A000041.

Programs

Mathematica

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];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
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]];
Reap[For[k = 0, k <= 1000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 17 2021 *)

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
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

Formula

A091891(a(n)) = 1.

Extensions

Terms a(40) and beyond from Andrew Howroyd, Apr 20 2021

a(1)=0 inserted by Alois P. Heinz, Dec 12 2021