A135560 -id:A135560 - cp's OEIS frontend

A135561 a(n) = 2^A135560(n) - 1.

3, 7, 1, 15, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 63, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 127, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 255, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 31, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 63, 1, 3

Offset: 1

N. J. A. Sloane, Mar 01 2008

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007814, A036987, A091090, A135534, A135560.

f[n_] := 1 + IntegerExponent[n, 2] + Sum[(-1)^(n - k - 1)*Binomial[n - 1, k]* Sum[Binomial[k, 2^j - 1], {j, 0, k}], {k, 0, n - 1}]; Table[2^f[k] - 1, {k, 1, 20}] (* G. C. Greubel, Oct 17 2016 *)

def A135561(n): return (1<<(m:=(~n & n-1)).bit_length()+int(m==n-1)+1)-1 # Chai Wah Wu, Jul 06 2022

a(2^k) = 2^(k+2) - 1; a(2^k + 2^(k-1)) = 2^k - 1. - Reinhard Zumkeller, Mar 02 2008

More terms from Reinhard Zumkeller, Mar 02 2008

A135523 a(n) = A007814(n) + A209229(n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A135416, A007814, A036987.
This is Guy Steele's sequence GS(4, 1) (see A135416).
One less than A135560.

Maple
```
GS(4,1,200); [see A135416].
```

A135523(n) = (valuation(n,2)+(n && !bitand(n,n-1))); \\ Antti Karttunen, Sep 27 2018

def A135523(n): return (~n& n-1).bit_length()+int(not(n&-n)^n) # Chai Wah Wu, Jul 08 2022

G.f.: x + Sum_{k>=1} x^(2^k)*(1 + 1/(1 - x^(2^k))). - Ilya Gutkovskiy, Mar 30 2017

a(n) = A135560(n) - 1. Antti Karttunen, Sep 27 2018

A135534 a(1) = 1; for n>=1, a(2n) = A135561(n), a(2n+1) = 0.

Original entry on oeis.org

1, 3, 0, 7, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 63, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 127, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A135416.

This is Guy Steele's sequence GS(6, 1) (see A135416).

Maple
```
GS(6,1,200); [see A135416].
```

A135560(n) = { my(t=valuation(n, 2)); (t + (n==2^t) + 1); }; \\ From A135560
A135534(n) = if(1==n,1,if((n%2),0,((2^(A135560(n/2)))-1))); \\ Antti Karttunen, Sep 27 2018

def A135534(n): return 1 if n == 1 else 0 if n&1 else (1<<(m:=(~(k:=n>>1) & k-1)).bit_length()+int(m==k-1)+1)-1 # Chai Wah Wu, Jul 06 2022

A193259 G.f.: x = Sum_{n>=1} x^n * ((1+x)^n - x^n) / (1+x)^a(n).

Original entry on oeis.org

1, 4, 5, 9, 10, 12, 13, 18, 19, 21, 22, 25, 26, 28, 29, 35, 36, 38, 39, 42, 43, 45, 46, 50, 51, 53, 54, 57, 58, 60, 61, 68, 69, 71, 72, 75, 76, 78, 79, 83, 84, 86, 87, 90, 91, 93, 94, 99, 100, 102, 103, 106, 107, 109, 110, 114, 115, 117, 118, 121, 122, 124, 125, 133, 134, 136, 137, 140, 141, 143, 144, 148, 149, 151, 152, 155, 156, 158, 159, 164, 165, 167

Offset: 1

Paul D. Hanna, Jul 19 2011

nonn

Sum_{n>=1} 1/phi^a(n) = 1/phi + Sum_{n>=1} 1/phi^(a(n) + 2*n) = 0.880771363850914609641... where phi = (sqrt(5)+1)/2. - Paul D. Hanna, Sep 14 2011

			G.f.: x = x*((1+x) - x)/(1+x) + x^2*((1+x)^2 - x^2)/(1+x)^4 + x^3*((1+x)^3 - x^3)/(1+x)^5 + x^4*((1+x)^4 - x^4)/(1+x)^9 + x^5*((1+x)^5 - x^5)/(1+x)^10 + x^6*((1+x)^6 - x^6)/(1+x)^12 + x^7*((1+x)^7 - x^7)/(1+x)^13 + x^8*((1+x)^8 - x^8)/(1+x)^18 +...+ x^n*((1+x)^n - x^n)/(1+x)^a(n) +...

Cf. A193263, A193260, A135560, A011371.

{a(n)=if(n<1,0,n + floor(log(n+1/2)/log(2)) + valuation(n!,2))}

{a(n)=if(n<1,0,if(n==1,1,polcoeff(sum(m=1,n+1,x^m*((1+x)^m-x^m)/(1+x +x^2*O(x^n))^if(m>=n,1,a(m)))+x^(n+1),n+1)))}

a(n) = n + floor(log_2(n)) + A011371(n) for n>=1, where A011371(n) = highest power of 2 dividing n!.

a(n) = a(n-1) + A135560(n) for n>1 with a(1)=1.

A241235 a(n) = number of times n appears in A006949.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 17 2014

nonn

Is this (with an exception at n=1) the same as A135560? - R. J. Mathar, Apr 26 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A051135.

a241235 n = a241235_list !! (n-1)
a241235_list = map length $ group a006949_list

cp's OEIS Frontend

A135561 a(n) = 2^A135560(n) - 1.

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A135523 a(n) = A007814(n) + A209229(n).

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A135534 a(1) = 1; for n>=1, a(2n) = A135561(n), a(2n+1) = 0.

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A193259 G.f.: x = Sum_{n>=1} x^n * ((1+x)^n - x^n) / (1+x)^a(n).

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A241235 a(n) = number of times n appears in A006949.

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