A091889 -id:A091889 - cp's OEIS frontend

A091890 Number of partitions of n into sums of exactly three distinct powers of 2.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 3, 2, 0, 1, 3, 3, 2, 5, 2, 1, 1, 5, 6, 3, 9, 5, 4, 5, 10, 9, 8, 13, 8, 10, 8, 16, 17, 15, 22, 18, 18, 20, 25, 28, 27, 34, 31, 32, 33, 44, 49, 44, 64, 53, 56, 61, 71, 77, 77, 100, 88, 94, 99, 123, 125, 132, 162, 147, 154

Offset: 1

Reinhard Zumkeller, Feb 10 2004

nonn

			a(14)=2: 14 = (2^3+2^2+2^1) = (2^2+2^1+2^0)+(2^2+2^1+2^0).

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A014311, A018819, A091889, A000120, A000041, A091893, A091891.

With[{max = 80}, m = Select[Range[max], DigitCount[#, 2, 1] == 3 &]; a[n_] := Length@ IntegerPartitions[n, n, m]; Array[a, max]] (* Amiram Eldar, Aug 01 2023 *)

A091891 Number of partitions of n into parts which are a sum of exactly as many distinct powers of 2 as n has 1's in its binary representation.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 2, 1, 10, 3, 2, 1, 5, 1, 2, 1, 36, 6, 12, 1, 11, 3, 2, 1, 24, 3, 3, 1, 5, 1, 2, 1, 202, 67, 55, 9, 93, 4, 5, 1, 112, 8, 13, 1, 10, 3, 2, 1, 304, 22, 18, 1, 20, 3, 3, 1, 34, 3, 3, 1, 5, 1, 2, 1, 1828, 1267, 1456, 71, 1629, 77, 100, 2, 2342, 99, 123, 9, 132, 4, 3, 1

Offset: 0

Reinhard Zumkeller, Feb 10 2004

			a(9) = 3 because there are 3 partitions of 9 into parts of size 3, 5, 6, 9 which are the numbers that have two 1's in their binary representations. The 3 partitions are: 9, 6 + 3 and 3 + 3 + 3. - _Andrew Howroyd_, Apr 20 2021

Alois P. Heinz, Table of n, a(n) for n = 0..16384 (terms n = 1..1000 from Andrew Howroyd)

Cf. A000079, A000120, A000041, A018819, A018900, A014311.

Cf. A091889, A091890, A091892, A091893.

H:= proc(n) option remember; add(i, i=Bits[Split](n)) end:
v:= proc(n, k) option remember; `if`(n<1, 0,
      `if`(H(n)=k, n, v(n-1, k)))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, v(i-1, k), k)+b(n-i, v(min(n-i, i), k), k)))
    end:
a:= n-> b(n$2, H(n)):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 12 2021

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]]];
a[n_] := EulerT[Table[DigitCount[k, 2, 1] == DigitCount[n, 2, 1] // Boole, {k, 1, n}]][[n]];
Array[a, 100] (* Jean-François Alcover, Dec 12 2021, after Andrew Howroyd *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
a(n) = {EulerT(vector(n,k,hammingweight(k)==hammingweight(n)))[n]} \\ Andrew Howroyd, Apr 20 2021

a(A000079(n)) = A018819(n);
a(A018900(n)) = A091889(n);
a(A014311(n)) = A091890(n);
a(A091892(n)) = 1.

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

A091893 Number of partitions of n into numbers having all the same number of 1's in binary representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 11, 13, 16, 17, 25, 23, 31, 35, 40, 42, 59, 53, 72, 79, 87, 91, 119, 114, 140, 151, 171, 178, 219, 203, 257, 275, 296, 319, 382, 364, 435, 467, 511, 533, 635, 609, 721, 764, 826, 865, 1011, 994, 1141, 1203, 1301, 1371, 1571, 1541, 1773

Offset: 1

Reinhard Zumkeller, Feb 10 2004

nonn

			n=30 -> '11110', a(30) = (#partitions into numbers with 1 binary 1) + (#partitions into numbers with 2 binary 1's) + (#partitions into numbers with 3 binary 1's) + (#partitions into numbers with 4 binary 1's) + (#partitions into numbers with 5 binary 1's) = A018819(30) + A091889(30) + A091890(30) + #{'11110','1111'+'1111'} + #empty = 166 + 50 + 1 + 2 + 0 = 219.

David A. Corneth, Table of n, a(n) for n = 1..10000

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

Table[Count[IntegerPartitions[n],?(Length[Union[DigitCount[ #,2,1]&/@ #]]==1&)],{n,60}] (* _Harvey P. Dale, Aug 23 2020 *)

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.

Original entry on oeis.org

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

Reinhard Zumkeller, Feb 10 2004

nonn

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

			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)

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)

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

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 *)

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

A091891(a(n)) = 1.

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

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

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A091890 Number of partitions of n into sums of exactly three distinct powers of 2.

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A091891 Number of partitions of n into parts which are a sum of exactly as many distinct powers of 2 as n has 1's in its binary representation.

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A091893 Number of partitions of n into numbers having all the same number of 1's in binary representation.

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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.

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