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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.
3
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
1,3
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
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)
FORMULA
A091891(a(n)) = 1.
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 10 2004
EXTENSIONS
Terms a(40) and beyond from Andrew Howroyd, Apr 20 2021
a(1)=0 inserted by Alois P. Heinz, Dec 12 2021
STATUS
approved