A023359 -id:A023359 - cp's OEIS frontend

A347784 Number of compositions (ordered partitions) of n into at most 3 powers of 2.

1, 1, 2, 3, 5, 5, 6, 6, 5, 5, 8, 6, 6, 6, 6, 0, 5, 5, 8, 6, 8, 6, 6, 0, 6, 6, 6, 0, 6, 0, 0, 0, 5, 5, 8, 6, 8, 6, 6, 0, 8, 6, 6, 0, 6, 0, 0, 0, 6, 6, 6, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 5, 5, 8, 6, 8, 6, 6, 0, 8, 6, 6, 0, 6, 0, 0, 0, 8, 6, 6, 0, 6, 0, 0, 0, 6

Offset: 0

Ilya Gutkovskiy, Sep 13 2021

nonn

Cf. A000079, A023359, A151759, A347785, A347786, A347787.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,3,Select[Range@n,IntegerQ@Log2@#&]],1],{n,0,100}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347785 Number of compositions (ordered partitions) of n into at most 4 powers of 2.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 12, 14, 18, 17, 18, 22, 24, 22, 26, 24, 18, 17, 26, 22, 30, 30, 34, 24, 24, 22, 34, 24, 26, 24, 24, 0, 18, 17, 26, 22, 38, 30, 34, 24, 30, 30, 42, 24, 34, 24, 24, 0, 24, 22, 34, 24, 34, 24, 24, 0, 26, 24, 24, 0, 24, 0, 0, 0, 18, 17, 26, 22, 38, 30

Offset: 0

Ilya Gutkovskiy, Sep 13 2021

nonn

Cf. A000079, A023359, A151760, A347784, A347786, A347787.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,4,Select[Range@n,IntegerQ@Log2@#&]],1],{n,0,69}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347786 Number of compositions (ordered partitions) of n into at most 5 powers of 2.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 24, 33, 42, 49, 52, 64, 72, 76, 84, 93, 82, 81, 92, 96, 100, 124, 124, 124, 112, 124, 124, 136, 124, 144, 120, 93, 82, 121, 92, 128, 140, 164, 124, 156, 140, 172, 164, 184, 164, 184, 120, 124, 112, 164, 124, 184, 164, 184, 120, 136, 124, 184, 120, 144

Offset: 0

Ilya Gutkovskiy, Sep 13 2021

nonn

Cf. A000079, A023359, A151761, A347784, A347785, A347787.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,5,Select[Range@n,IntegerQ@Log2@#&]],1],{n,0,60}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347787 Number of compositions (ordered partitions) of n into at most 6 powers of 2.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 30, 48, 68, 94, 118, 140, 168, 202, 224, 258, 292, 302, 302, 336, 352, 370, 424, 470, 472, 490, 504, 532, 544, 584, 600, 618, 532, 526, 542, 544, 536, 674, 664, 666, 656, 754, 704, 820, 824, 904, 840, 830, 712, 794, 744, 820, 824, 984

Offset: 0

Ilya Gutkovskiy, Sep 13 2021

nonn

Cf. A000079, A023359, A151762, A347784, A347785, A347786.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,6,Select[Range@n,IntegerQ@Log2@#&]],1],{n,0,54}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A050379 Number of ordered factorizations of n into members of A050376.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 6, 1, 5, 1, 5, 2, 2, 1, 10, 2, 2, 3, 5, 1, 6, 1, 10, 2, 2, 2, 14, 1, 2, 2, 10, 1, 6, 1, 5, 5, 2, 1, 22, 2, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 18, 1, 2, 5, 18, 2, 6, 1, 5, 2, 6, 1, 32, 1, 2, 5, 5, 2, 6, 1, 22, 6, 2, 1, 18, 2, 2, 2, 10, 1, 18, 2, 5, 2, 2, 2

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A002033, A050376, A050377, A050378, A050380.

Cf. also A000142, A002110, A023359.

read(transforms) :
L := [1] :
for n from 2 to 100  do
    if isA050376(n) then
        L := [op(L),-1] ;
    else
        L := [op(L),0] ;
    end if;
end do :
a050379 := DIRICHLETi(L) ; # R. J. Mathar, May 26 2017

A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus, Feb 10 2016
isA050376(n) = ((1==omega(n)) && (1==A064547(n))); \\ Checking that omega(n) is 1 is just an optimization here.
A050379(n) = if(1==n,n,sumdiv(n,d,if(dA050376(n/d)*A050379(d),0))); \\ Antti Karttunen, Oct 20 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of A050376.
a(p^k) = A023359(k), for any prime p.
a(A002110(n)) = A000142(n) = n!.
a(n) = A050380(A101296(n)). - R. J. Mathar, May 26 2017

A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2k-1))x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 16, 21, 26, 37, 47, 61, 84, 108, 143, 191, 249, 331, 437, 575, 763, 1004, 1326, 1754, 2311, 3055, 4036, 5323, 7033, 9288, 12257, 16193, 21379, 28223, 37278, 49212, 64984, 85815, 113297, 149614, 197551, 260839, 344439, 454795, 600517, 792958, 1047023, 1382519, 1825533, 2410456, 3182845

Offset: 0

Ilya Gutkovskiy, Dec 28 2016

nonn

Number of compositions (ordered partitions) into odd prime powers (1 excluded).

			a(10) = 3 because we have [7, 3], [5, 5] and [3, 7].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to compositions

Cf. A001221, A023359, A061345, A246655, A280151, A280195.

nmax = 60; CoefficientList[Series[1/(1 - Sum[Floor[1/PrimeNu[2 k - 1]] x^(2 k - 1), {k, 2, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} floor(1/omega(2*k-1))*x^(2*k-1)).

A281811 Expansion of Sum_{i>=0} x^(2^i) / (1 - Sum_{j>=0} x^(2^j))^2.

Original entry on oeis.org

1, 3, 7, 16, 34, 71, 143, 286, 562, 1096, 2114, 4054, 7720, 14631, 27591, 51834, 97018, 181030, 336810, 625062, 1157288, 2138200, 3942858, 7257830, 13338024, 24474978, 44848232, 82073852, 150016328, 273893503, 499534495, 910161570, 1656786466, 3013237398, 5475710770, 9942780954, 18040712384, 32711070838

Offset: 1

Ilya Gutkovskiy, Jan 30 2017

nonn

Total number of parts in all compositions (ordered partitions) of n into powers of 2 (A000079).

			a(4) = 16 because we have [4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 1 + 2 + 3 + 3 + 3 + 4 = 16.

Vaclav Kotesovec, Table of n, a(n) for n = 1..4000
Index entries for sequences related to compositions

Cf. A000079, A023359.

b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+[0, p[1]])(b(n-2^j)), j=0..ilog2(n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Aug 07 2019

nmax = 38; Rest[CoefficientList[Series[Sum[x^2^i, {i, 0, nmax}]/(1 - Sum[x^2^j, {j, 0, nmax}])^2, {x, 0, nmax}], x]]
nmax = 40; Rest[CoefficientList[Series[Sum[x^(2^i), {i, 0, Floor[Log[nmax]/Log[2]] + 1}]/(1 - Sum[x^(2^j), {j, 0, Floor[Log[nmax]/Log[2]] + 1}])^2, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Feb 17 2017 *)

G.f.: Sum_{i>=0} x^(2^i) / (1 - Sum_{j>=0} x^(2^j))^2.

a(n) ~ c * n / r^n, where r = 0.566123792684559918241681653033264449147... is the root of the equation Sum_{j>=0} r^(2^j) = 1 and c = 0.34432689951558638915900387175922521737229978512101795819134... . - Vaclav Kotesovec, Feb 17 2017

A382187 Expansion of 1/(1 - 4 * Sum_{k>=0} x^(2^k))^(1/2).

Original entry on oeis.org

1, 2, 8, 32, 138, 604, 2696, 12176, 55512, 254888, 1177064, 5461040, 25435296, 118856272, 556962928, 2616287392, 12315914698, 58084552572, 274395134600, 1298187523792, 6150051540460, 29170558879736, 138512004786624, 658362443599296, 3132140164624680

Offset: 0

Seiichi Manyama, Mar 18 2025

Cf. A023359, A382188.

Cf. A223142.

G.f. A(x) satisfies A(x) = 1/(1/A(x^2)^2 - 4*x)^(1/2).

A382188 Expansion of 1/(1 - 9 * Sum_{k>=0} x^(2^k))^(1/3).

Original entry on oeis.org

1, 3, 21, 162, 1344, 11565, 102033, 916002, 8330331, 76515363, 708379137, 6600436794, 61829064882, 581783753232, 5495344743924, 52079440119336, 494985533135250, 4716537209764020, 45043670723519952, 431041661857081656, 4132290587464466820, 39680088682182010749

Offset: 0

Seiichi Manyama, Mar 18 2025

Cf. A023359, A382187.

G.f. A(x) satisfies A(x) = 1/(1/A(x^2)^3 - 9*x)^(1/3).

A109696 Decimal expansion of root of 1 - Sum_{n>=0} 1/x^(2^n).

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Aug 07 2005

			1.766398114550173597228488392440099730232069287957072775...

This is the limit ratio between consecutive terms of A023359.

RealDigits[ FindRoot[1 - Sum[1/(x^(2^n)), {n, 0, 8}] == 0, {x, 1.7}, WorkingPrecision -> 128][[1, 2]], 10, 128][[1]] (* Robert G. Wilson v, Aug 08 2005 *)

PARI
```
solve(x=1,2,1-sum(k=0,8,1./x^(2^k)))
```

cp's OEIS Frontend

A347784 Number of compositions (ordered partitions) of n into at most 3 powers of 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A347785 Number of compositions (ordered partitions) of n into at most 4 powers of 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A347786 Number of compositions (ordered partitions) of n into at most 5 powers of 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A347787 Number of compositions (ordered partitions) of n into at most 6 powers of 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A050379 Number of ordered factorizations of n into members of A050376.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2*k-1))*x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A281811 Expansion of Sum_{i>=0} x^(2^i) / (1 - Sum_{j>=0} x^(2^j))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A382187 Expansion of 1/(1 - 4 * Sum_{k>=0} x^(2^k))^(1/2).

Views

Author

Keywords

Crossrefs

Formula

A382188 Expansion of 1/(1 - 9 * Sum_{k>=0} x^(2^k))^(1/3).

Views

Author

Keywords

Crossrefs

Formula

A109696 Decimal expansion of root of 1 - Sum_{n>=0} 1/x^(2^n).

Views

Author

Keywords

Examples

A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2k-1))x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).