A007731 -id:A007731 - cp's OEIS frontend

A083662 a(n) = a(floor(n/2)) + a(floor(n/4)), n > 0; a(0)=1.

1, 2, 3, 3, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34

Offset: 0

Benoit Cloitre, Oct 05 2003

nonn

A000045(n+2) = a(A131577(n))and A000045(m+2) < a(m) for m < A131577(n). - Reinhard Zumkeller, Sep 26 2009

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A088468.

A007731, A165704, A165706. - Reinhard Zumkeller, Sep 26 2009

PARI
```
a(n)=if(n<1,n==0,a(n\2)+a(n\4))
```

For n > 0, a(n) = F([log(n)/log(2)]+3) where F(k) denotes the k-th Fibonacci number. For n >= 3, F(n) appears 2^(n-3) times. More generally, if p is an integer > 1 and a(n) = a(floor(n/p)) + a(floor(n/p^2)), n > 0, a(0)=1, then for n > 0, a(n) = F(floor(log(n)/log(p)) + 3).

A088468 a(0) = 1, a(n) = a(floor(n/2)) + a(floor(n/3)) for n > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 7, 8, 9, 9, 9, 12, 12, 12, 12, 13, 13, 16, 16, 16, 16, 16, 16, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 33, 33, 33, 33, 33, 33, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 38, 38, 38, 48

Offset: 0

Michael Somos, Oct 02 2003

nonn

Record values greater than 1 occur at 3-smooth numbers: A160519(n)=a(A003586(n)) and A160519(m)A003586(n). - Reinhard Zumkeller, May 16 2009

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
A. R. Lebeck, CPS 104: Homework #3.
Michael Penn, Erdős but simpler, Youtube video.

Equals A061984(n) + 1.

Cf. A007731, A083662, A165704, A165706. - Reinhard Zumkeller, Sep 26 2009

a[0]=1;a[n_]:=a[n]=a[Floor[n/2]]+a[Floor[n/3]];Array[a,75,0] (* Harvey P. Dale, Aug 23 2020 *)

PARI
```
a(n)=if(n<1,n==0,a(n\2)+a(n\3))
```

Limit_{n->oo} a(n)/n = 0, as proved in Michael Penn's Youtube video (see Links). Michael Penn states in the video that this is a simplification of a problem of Paul Erdős, where the original problem is to show that limit_{n->oo} b(n)/n = 12/log(432) for b(0) = 1, b(n) = b(floor(n/2)) + b(floor(n/3)) + b(floor(n/6)) for n > 0 ({b(n)} is the sequence A007731). - Jianing Song, Sep 27 2023

A061984 a(n) = 1 + a([n/2]) + a([n/3]) with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 6, 6, 7, 8, 8, 8, 11, 11, 11, 11, 12, 12, 15, 15, 15, 15, 15, 15, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 32, 32, 32, 32, 32, 32, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 37, 37, 37, 37, 37, 37, 37, 37, 47

Offset: 0

Henry Bottomley, May 24 2001

nonn

If n = 2^a*3^b, then a(n)-a(n-1) = C(a+b, a). - David Wasserman, Nov 17 2005

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A061980, A061985, A061986, A061987.

Cf. A007731.

a061984 n = a061984_list !! n
a061984_list = 0 : map (+ 1) (zipWith (+)
   (map (a061984 . (`div` 2)) [1..]) (map (a061984 . (`div` 3)) [1..]))
-- Reinhard Zumkeller, Jan 11 2014

A165704 a(n) = a([n/2]) + a([n/3]) + a([n/5]).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 15, 17, 19, 23, 23, 29, 29, 29, 33, 35, 35, 41, 41, 47, 47, 47, 47, 55, 57, 57, 59, 59, 59, 71, 71, 73, 73, 73, 73, 85, 85, 85, 85, 93, 93, 93, 93, 93, 99, 99, 99, 109, 109, 115, 115, 115, 115, 123, 123, 123, 123, 123, 123, 147, 147, 147, 147, 149

Offset: 0

Reinhard Zumkeller, Sep 26 2009

nonn

Record values greater than 1 occur at 5-smooth numbers, n>0: A165705(n)=a(A051037(n)) and A165705(m)A051037(n).

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007731, A083662, A088468, A165706.

A165706 a(0) = 1, a(n) = a([n/2]) + a([n/5]) for n > 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22

Offset: 0

Reinhard Zumkeller, Sep 26 2009

nonn

For n>0: A165707(n)=a(A003592(n)) and A165707(m)A003592(n).

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007731, A083662, A088468, A165704.

a(n)=if(n<3, return(n+1)); a(n\2) + a(n\5) \\ Charles R Greathouse IV, Nov 14 2016

A098971 a(0)=1; for n > 0, a(n)=a(floor(n/2))+2*a(floor(n/4)).

Original entry on oeis.org

1, 3, 5, 5, 11, 11, 11, 11, 21, 21, 21, 21, 21, 21, 21, 21, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 171, 171, 171, 171, 171

Offset: 0

Benoit Cloitre, Oct 23 2004

nonn

a(2^n) gives the Jacobsthal sequence A001045(n+3).

Cf. A001045, A007731.

a(n)=if(n<1,1,(1/3)*(8*2^(floor(log(n)/log(2)))+(-1)^(floor(log(n)/log(2)))))

def A098971(n): return ((1<<(m:=n.bit_length()+2))+(1 if m&1 else -1))//3 # Chai Wah Wu, Oct 10 2024

n>0, a(n) = (1/3)*(8*2^(floor(log(n)/log(2)))+(-1)^(floor(log(n)/log(2)))).

A098972 a(0) = 1; for n > 0, a(n) = 2a(floor(n/2)) + 3a(floor(n/3)).

Original entry on oeis.org

1, 5, 13, 25, 41, 41, 89, 89, 121, 157, 157, 157, 301, 301, 301, 301, 365, 365, 581, 581, 581, 581, 581, 581, 965, 965, 965, 1073, 1073, 1073, 1073, 1073, 1201, 1201, 1201, 1201, 2065, 2065, 2065, 2065, 2065, 2065, 2065, 2065, 2065, 2065, 2065, 2065, 3025

Offset: 0

Benoit Cloitre, Oct 23 2004

nonn

a(n) > a(n-1) iff n is a 3-smooth number.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A003586, A007731, A065333.

a[0] = 1; a[n_] := a[n] = 2*a[Floor[n/2]] + 3*a[Floor[n/3]]; Array[a, 50, 0] (* Amiram Eldar, Jul 13 2023 *)

a(n)=if(n<1,1,a(floor(n/2))*2+3*a(floor(n/3)))

sign(a(n+1)-a(n)) = A065333(n+1).

cp's OEIS Frontend

A083662 a(n) = a(floor(n/2)) + a(floor(n/4)), n > 0; a(0)=1.

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Formula

A088468 a(0) = 1, a(n) = a(floor(n/2)) + a(floor(n/3)) for n > 0.

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Mathematica

PARI

Formula

A061984 a(n) = 1 + a([n/2]) + a([n/3]) with a(0) = 0.

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Haskell

A165704 a(n) = a([n/2]) + a([n/3]) + a([n/5]).

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A165706 a(0) = 1, a(n) = a([n/2]) + a([n/5]) for n > 1.

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PARI

A098971 a(0)=1; for n > 0, a(n)=a(floor(n/2))+2*a(floor(n/4)).

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Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A098972 a(0) = 1; for n > 0, a(n) = 2*a(floor(n/2)) + 3*a(floor(n/3)).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A098972 a(0) = 1; for n > 0, a(n) = 2a(floor(n/2)) + 3a(floor(n/3)).