A018900 -id:A018900 - cp's OEIS frontend

A038461 Sums of 10 distinct powers of 2.

1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 2046, 2559, 2815, 2943, 3007, 3039, 3055, 3063, 3067, 3069, 3070, 3327, 3455, 3519, 3551, 3567, 3575, 3579, 3581, 3582, 3711, 3775, 3807, 3823, 3831, 3835, 3837, 3838, 3903

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Base 2 interpretation of A038452.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9).

Select[Range[4000], DigitCount[#, 2, 1] == 10 &] (* Amiram Eldar, Feb 14 2022 *)

isok(n) = hammingweight(n) == 10; \\ Michel Marcus, Feb 29 2016

from itertools import islice
def A038461_gen(): # generator of terms
    yield (n:=1023)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A038461_list = list(islice(A038461_gen(),20)) # Chai Wah Wu, Mar 10 2025

Sum_{n>=1} 1/a(n) = 1.386312271262110321181505974797071257205562524228381227122302929089588534920... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A091889 Number of partitions of n into sums of exactly two distinct powers of 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 2, 2, 5, 2, 3, 8, 4, 6, 12, 6, 11, 16, 11, 16, 24, 17, 23, 34, 26, 35, 50, 35, 50, 67, 55, 72, 93, 76, 99, 126, 112, 135, 171, 150, 186, 229, 210, 249, 304, 280, 336, 398, 380, 443, 526, 499, 584, 680, 665, 759, 886, 858, 985, 1136, 1123

Offset: 1

Reinhard Zumkeller, Feb 10 2004

nonn

			a(9)=3: 9 = (2^3+2^0) = (2^2+2^1)+(2^1+2^0) = (2^1+2^0)+(2^1+2^0)+(2^1+2^0).

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

Cf. A018819, A018900, A091890, A000120, A000041, A091893, A091891.

With[{max = 64}, m = Select[Range[max], DigitCount[#, 2, 1] == 2 &]; 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

A092429 a(n) = n! * Sum_{i,j,k,l >= 0, i+j+k+l = n} 1/(i!j!k!*l!).

Original entry on oeis.org

1, 1, 3, 10, 47, 126, 522, 1821, 8143, 26326, 109958, 396111, 1737122, 5998955, 24949277, 91979985, 397402223, 1418993350, 5881338702, 22010456331, 94022106862, 342803313261, 1416758002487, 5356198979731, 22685035586290, 83911052895151, 345921828889367

Offset: 0

Benoit Cloitre, Mar 22 2004

nonn

a(n) is even iff n is a sum of 2 distinct powers of 2.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence (of order 11)

Cf. A018900, A027306, A092255.

Column k=4 of A226873. - Alois P. Heinz, Jun 21 2013

b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 4):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 21 2017

Table[Sum[Sum[Sum[Sum[If[i+j+k+l==n,n!/i!/j!/k!/l!,0],{l,0,k}],{k,0,j}],{j,0,i}],{i,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 01 2013 *)
CoefficientList[Series[(HypergeometricPFQ[{},{},x]^4 +6*HypergeometricPFQ[{},{},x]^2 *HypergeometricPFQ[{},{1},x^2] +8*HypergeometricPFQ[{},{},x] *HypergeometricPFQ[{},{1,1},x^3] +3*HypergeometricPFQ[{},{1},x^2]^2 +6*HypergeometricPFQ[{},{1,1,1},x^4])/24, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec after Vladeta Jovovic, Jul 01 2013 *)

a(n)=sum(i=0,n,sum(j=0,i,sum(k=0,j,sum(l=0,k,if(i+j+k+l-n,0,n!/i!/j!/k!/l!)))))

E.g.f.: (t(1)^4 + 6*t(1)^2*t(2) + 8*t(1)*t(3) + 3*t(2)^2 + 6*t(4))/24 where t(1) = hypergeom([],[],x), t(2) = hypergeom([],[1],x^2), t(3) = hypergeom([],[1,1],x^3) and t(4) = hypergeom([],[1,1,1],x^4). - Vladeta Jovovic, Sep 22 2007, typo corrected by Vaclav Kotesovec, Jul 01 2013

Conjecture: a(n) ~ 4^n/4!. - Vaclav Kotesovec, Mar 07 2014

A110202 a(n) = sum of squares of numbers < 2^n having exactly 2 ones in their binary representation.

Original entry on oeis.org

0, 9, 70, 395, 1984, 9429, 43434, 196095, 872788, 3842729, 16773118, 72693075, 313158312, 1342144509, 5726557522, 24338016935, 103078952956, 435222828369, 1832518331046, 7696579297275, 32252336887120, 134873417951909

Offset: 1

Paul D. Hanna, Jul 16 2005

nonn

Equals column 2 of triangle A110200.

			For n=4, the sum of the squares of numbers < 2^4
having exactly 2 ones in their binary digits is:
a(4) = 3^2 + 5^2 + 6^2 + 9^2 + 10^2 + 12^2 = 395.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A110200 (triangle), A110201 (central terms), A002450 (column 1), A110203 (column 3), A110204 (column 4), A018900.

nn=30;With[{c=Union[FromDigits[#,2]&/@(Flatten[Table[Join[ {1},#]&/@ Permutations[Join[{1},PadRight[{},n,0]]],{n,0,nn}],1])]}, Table[ Total[ Select[c,#<2^n&]^2],{n,nn}]] (* Harvey P. Dale, Jan 27 2013 *)

a(n)=polcoeff(x^2*(9-38*x+32*x^2)/((1-x)^2*(1-2*x)*(1-4*x)^2+x*O(x^n)),n)

G.f.: x^2*(9-38*x+32*x^2)/( (1-x)^2*(1-2*x)*(1-4*x)^2 ). a(n) = Sum_{k|A018900(k)<2^n} A018900(k)^2.

A151758 G.f.: Theta^2-Theta, where Theta = Sum_{k>=0} x^(2^k).

Original entry on oeis.org

0, -1, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jun 22 2009

sign

If we omit the "-x" term and divide by 2, we get the characteristic function of the numbers with binary weight 2 (A018900, A151774).

Cf. A000079, A073267, A151759, A151760, A151761, A151762, A018900, A151774.

w[n_] := IntegerDigits[n, 2] // Total;
a[n_] := If[n == 1, -1, 2 Boole[w[n] == 2]];
a /@ Range[0, 104] (* Jean-François Alcover, Mar 31 2021 *)

A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

Original entry on oeis.org

1, 5, 2, 8, 9, 9, 9, 5, 6, 0, 6, 9, 6, 8, 8, 8, 4, 1, 8, 3, 8, 2, 6, 3, 9, 4, 9, 4, 5, 1, 0, 9, 9, 6, 9, 6, 5, 1, 1, 5, 3, 9, 3, 9, 9, 7, 7, 1, 5, 0, 5, 1, 2, 5, 3, 1, 3, 2, 4, 7, 5, 9, 2, 0, 5, 3, 1, 7, 5, 1, 3, 5, 9, 5, 3, 2, 0, 1, 4, 1, 7, 0, 1, 2, 3, 8, 0, 8, 8, 6, 4, 3, 0, 5, 7, 0, 7, 9, 7, 0, 2, 2, 2, 7, 0

Offset: 1

Robert G. Wilson v, Aug 03 2010

Obviously for k > 0 in base 2 having no bit equal to 1 the sum is 0 and for 1 bit equal to 1 the sum is 2.

			Sum_{k>0} 1/A018900(k) = 1.52899956069688841838263949451...

Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Cf. A065442, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839, A140502, A160502, A018900, A323482.

 evalf( 2*add( (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2), n = 1..18), 100); # Peter Bala, Jan 28 2022

(* first install irwinSums.m, see either reference, then *) First@ RealDigits@ iSum[1, 2, 2^7, 2]

Equals Sum_{j>=1} Sum_{i=0..j-1} 1/(2^i + 2^j).
From Amiram Eldar, Jun 30 2020: (Start)
Equals Sum_{k>=0} 1/(2^k + 1/2).
Equals 2 * A323482 - 1. (End)
Equals 2*Sum_{n >= 1} (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2). The first 18 terms of the series gives the constant correct to more than 100 decimal places. - Peter Bala, Jan 28 2022

A038462 Sums of 11 distinct powers of 2.

Original entry on oeis.org

2047, 3071, 3583, 3839, 3967, 4031, 4063, 4079, 4087, 4091, 4093, 4094, 5119, 5631, 5887, 6015, 6079, 6111, 6127, 6135, 6139, 6141, 6142, 6655, 6911, 7039, 7103, 7135, 7151, 7159, 7163, 7165, 7166, 7423, 7551, 7615, 7647, 7663, 7671

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Base 2 interpretation of A038453.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461 (Hamming weight = 1, 2, ..., 10).

Select[Range[8000], DigitCount[#, 2, 1] == 11 &] (* Amiram Eldar, Feb 14 2022 *)

from itertools import islice
def A038462_gen(): # generator of terms
    yield (n:=2047)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A038462_list = list(islice(A038462_gen(),20)) # Chai Wah Wu, Mar 10 2025

Sum_{n>=1} 1/a(n) = 1.386300330514503033229968047555778179200262625510401687087371496738972082061... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A211362 Inversion sets of finite permutations interpreted as binary numbers.

Original entry on oeis.org

0, 1, 4, 3, 6, 7, 32, 33, 20, 11, 22, 15, 48, 41, 52, 43, 30, 31, 56, 57, 60, 59, 62, 63, 512, 513, 516, 515, 518, 519, 288, 289, 148, 75, 150, 79, 304, 297, 180, 107, 158, 95, 312, 313, 188, 123, 190, 127, 768, 769, 644, 579, 646, 583, 800

Offset: 0

Tilman Piesk, Jun 03 2012

nonn

Each finite permutation has a finite inversion set. The possible elements of the inversion sets are 2-element subsets of the integers, which can be ordered in an infinite sequence (compare A018900). Thus the inversion set can be represented by a binary vector, which can be interpreted as a binary number.
This sequence shows these numbers for the finite permutations in reverse colexicographic order (A055089, A195663). The corresponding inversion vectors are found in A007623. The corresponding inversion numbers (A034968) are the digit sums of the inversion vectors and the cardinality of the inversion sets, an thus also the binary digit sums of the numbers in this sequence.
This sequence is not monotonic. The permutation A211363 shows how the elements of this sequence (a) are ordered. a*A211363 gives the elements of a ordered by size.

			The 4th finite permutation (2,3,1,4,...) has the inversion set {(1,3),(2,3)}. This set represented by a vector is (0,1,1,zeros...). This vector interpreted as a number is 6. So a(4)=6.
The 23rd finite permutation (4,3,2,1,...) has the inversion set {(1,2),(1,3),(2,3),(1,4),(2,4),(3,4)}. This set represented by a vector is (1,1,1,1,1,1,zeros...). This vector interpreted as a number is 63. So a(23)=63.
Beginning of corresponding array:
n    permutation   inversion set    a(n)
00     1 2 3 4     0  0 0  0 0 0     0
01     2 1 3 4     1  0 0  0 0 0     1
02     1 3 2 4     0  0 1  0 0 0     4
03     3 1 2 4     1  1 0  0 0 0     3
04     2 3 1 4     0  1 1  0 0 0     6
05     3 2 1 4     1  1 1  0 0 0     7
06     1 2 4 3     0  0 0  0 0 1    32
07     2 1 4 3     1  0 0  0 0 1    33
08     1 4 2 3     0  0 1  0 1 0    20
09     4 1 2 3     1  1 0  1 0 0    11
10     2 4 1 3     0  1 1  0 1 0    22
11     4 2 1 3     1  1 1  1 0 0    15
12     1 3 4 2     0  0 0  0 1 1    48
13     3 1 4 2     1  0 0  1 0 1    41
14     1 4 3 2     0  0 1  0 1 1    52
15     4 1 3 2     1  1 0  1 0 1    43
16     3 4 1 2     0  1 1  1 1 0    30
17     4 3 1 2     1  1 1  1 1 0    31
18     2 3 4 1     0  0 0  1 1 1    56
19     3 2 4 1     1  0 0  1 1 1    57
20     2 4 3 1     0  0 1  1 1 1    60
21     4 2 3 1     1  1 0  1 1 1    59
22     3 4 2 1     0  1 1  1 1 1    62
23     4 3 2 1     1  1 1  1 1 1    63

Tilman Piesk, Table of n, a(n) for n = 0..5039
Wikipedia, Inversion (discrete mathematics)

Cf. A018900, A055089, A195663, A034968, A211363.

A337758 G.f. A(x) satisfies: [x^n] (1 + nx + nx^2 - A(x))^(n+1) = 0, for n > 0.

Original entry on oeis.org

1, 3, 8, 41, 284, 2594, 29420, 395845, 6109724, 105714438, 2017696504, 41979555034, 943466064072, 22739452659420, 584304270694436, 15928490898945133, 458761105965272316, 13910124960218668430, 442657291681105692624, 14744175994124292681518, 512800784035081173166088

Offset: 1

Paul D. Hanna, Oct 24 2020

nonn

Compare to: [x^n] (1 + n*x - C(x))^(n+1) = 0, for n>0, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Compare to: [x^n] (1 + n*x - W(x))^n = 0, for n>0, where W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n! = 1 + x/LambertW(-x).

			G.f.: A(x) = x + 3*x^2 + 8*x^3 + 41*x^4 + 284*x^5 + 2594*x^6 + 29420*x^7 + 395845*x^8 + 6109724*x^9 + 105714438*x^10 + 2017696504*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1 + n*x + n*x^2 - A(x))^(n+1) begins:
n=0: [1, -1, -3, -8, -41, -284, -2594, -29420, -395845, ...];
n=1: [1, 0, -4, -16, -78, -536, -4960, -57048, -775089, ...];
n=2: [1, 3, 0, -29, -171, -1071, -9124, -100590, -1334241, ...];
n=3: [1, 8, 24, 0, -340, -2504, -19032, -189408, -2368430, ...];
n=4: [1, 15, 95, 290, 0, -5327, -46335, -409770, -4606315, ...];
n=5: [1, 24, 252, 1472, 4614, 0, -103528, -1028952, -10296567, ...];
n=6: [1, 35, 546, 4949, 27972, 90244, 0, -2388773, -26537259, ...];
n=7: [1, 48, 1040, 13376, 112344, 627280, 2083504, 0, -63579020, ...];
n=8: [1, 63, 1809, 31260, 360765, 2901258, 16172964, 55276020, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x + n*x^2 - A(x))^(n+1) = 0, for n > 0.
ODD TERMS.
The odd terms seem to occur only at positions equal to powers of 2: a(1), a(2), a(4), a(8), a(16), ...; the odd terms begin: [1, 3, 41, 395845, 15928490898945133, 490833755530209698774408313021523960879357, ...].
RELATED SERIES.
B(x) = 1/(1 - A(x)) = 1 + x + 4*x^2 + 15*x^3 + 76*x^4 + 478*x^5 + 3868*x^6 + 39675*x^7 + 498120*x^8 + 7351430*x^9 + 123503516*x^10 + 2309531318*x^11 + ...
where [x^n] (1 + n*(x + x^2)*B(x))^(n+1) / B(x)^(n+1) = 0 for n > 0.
Series_Reversion(A(x)) = x - 3*x^2 + 10*x^3 - 56*x^4 + 268*x^5 - 2104*x^6 + 7636*x^7 - 129976*x^8 - 369988*x^9 - 19147364*x^10 - 279267684*x^11 - ...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A338328, A000108, A018900.

{a(n) = my(A=[1],m=1); for(i=1,n, A=concat(A,0);
m=#A; A[#A] = polcoeff( (1 + m*x + m*x^2 - x*Ser(A))^(m+1), m)/(m+1) );A[n]}
for(n=1,30,print1(a(n),", "))

a(n) is odd iff n is a power of 2 (conjecture).

a(n) = 2 (mod 4) iff n is twice the sum of two distinct powers of 2 (conjecture).

cp's OEIS Frontend

A038461 Sums of 10 distinct powers of 2.

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A091889 Number of partitions of n into sums of exactly two 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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A092429 a(n) = n! * Sum_{i,j,k,l >= 0, i+j+k+l = n} 1/(i!*j!*k!*l!).

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A110202 a(n) = sum of squares of numbers < 2^n having exactly 2 ones in their binary representation.

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A151758 G.f.: Theta^2-Theta, where Theta = Sum_{k>=0} x^(2^k).

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A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

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A038462 Sums of 11 distinct powers of 2.

A092429 a(n) = n! * Sum_{i,j,k,l >= 0, i+j+k+l = n} 1/(i!j!k!*l!).

A337758 G.f. A(x) satisfies: [x^n] (1 + nx + nx^2 - A(x))^(n+1) = 0, for n > 0.