A036442 -id:A036442 - cp's OEIS frontend

A344534 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+A002262(e_k))^2^A025581(e_k) (where prime(k) denotes the k-th prime number).

1, 2, 4, 8, 3, 6, 12, 24, 16, 32, 64, 128, 48, 96, 192, 384, 9, 18, 36, 72, 27, 54, 108, 216, 144, 288, 576, 1152, 432, 864, 1728, 3456, 5, 10, 20, 40, 15, 30, 60, 120, 80, 160, 320, 640, 240, 480, 960, 1920, 45, 90, 180, 360, 135, 270, 540, 1080, 720, 1440

Offset: 0

Rémy Sigrist, May 22 2021

The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).
The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:
    ...
      7| 9
      5| 5 8
      3| 2 4 7
      2| 0 1 3 6
    ---+--------
    p/k| 0 1 2 3 ...
This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344536.
This sequence establishes a bijection from A261195 to A225547.
This sequence and A344535 each map between two useful choices for encoding sets of elements drawn from a 2-dimensional array. To give a very specific example, each mapping is an isomorphism between two alternative integer representations of the polynomial ring GF2[x,y]. The relevant set is {x^i*y^j : i, j >= 0}. The mappings between the two representations of the ring's addition operation are from XOR (A003987) to A059897(.,.) and for the multiplication operation, they are from A329331(.,.) to A329329(.,.). - Peter Munn, May 31 2021

			For n = 42:
- 42 = 2^5 + 2^3 + 2^1,
- so we have the following Fermi-Dirac factors p^2^k:
      5| X
      3|
      2|   X X
    ---+------
    p/k| 0 1 2
- a(42) = 2^2^1 * 2^2^2 * 5^2^0 = 320.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Encyclopedia of Mathematics, Isomorphism
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A002262, A006125, A025581, A036442, A064547, A225546, A225547, A261195, A329050, A344531, A344535, A344536.
Comparable mappings that also use Fermi-Dirac factors: A052330, A059900.
Maps binary operations A003987 to A059897, A329331 to A329329.

A002262(n)=n-binomial(round(sqrt(2+2*n)), 2)
A025581(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1)
a(n) = { my (v=1, e); while (n, n-=2^e=valuation(n, 2); v* = prime(1 + A002262(e))^2^A025581(e)); v }

a(n) = A344535(A344531(n)).
a(n) = A344535(n) iff n belongs to A261195.
A064547(a(n)) = A000120(n).
a(A036442(n)) = prime(n).
a(A006125(n+1)) = 2^2^n for any n >= 0.
a(m + n) = a(m) * a(n) when m AND n = 0 (where AND denotes the bitwise AND operator).
From Peter Munn, Jun 06 2021: (Start)
a(n) = A225546(A344535(n)).
a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
a(A329331(n, k)) = A329329(a(n), a(k)).
(End)

A344535 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+A025581(e_k))^2^A002262(e_k) (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 16, 32, 48, 96, 64, 128, 192, 384, 80, 160, 240, 480, 320, 640, 960, 1920, 144, 288, 432, 864, 576, 1152, 1728, 3456, 720, 1440

Offset: 0

Rémy Sigrist, May 22 2021

The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).
The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:
    ...
      7| 6
      5| 3 7
      3| 1 4 8
      2| 0 2 5 9
    ---+--------
    p/k| 0 1 2 3 ...
This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344537.
This sequence establishes a bijection from A261195 to A225547.

			For n = 42:
- 42 = 2^5 + 2^3 + 2^1,
- so we have the following Fermi-Dirac factors p^2^k:
      5| X
      3| X
      2|     X
    ---+------
    p/k| 0 1 2
- a(42) = 3^2^0 * 5^2^0 * 2^2^2 = 240.

Cf. A000120, A002262, A006125, A025581, A036442, A052330, A064547, A225547, A261195, A344531, A344534, A344537.

A002262(n)=n-binomial(round(sqrt(2+2*n)), 2)
A025581(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1)
a(n) = { my (v=1, e); while (n, n-=2^e=valuation(n, 2); v *= prime(1 + A025581(e))^2^A002262(e)); v }

a(n) = A344534(A344531(n)).
a(n) = A344534(n) iff n belongs to A261195.
A064547(a(n)) = A000120(n).
a(A006125(n)) = prime(n) for any n > 0.
a(A036442(n+1)) = 2^2^n for any n >= 0.
a(m + n) = a(m) * a(n) when m AND n = 0 (where AND denotes the bitwise AND operator).

A080609 Binomial transform of central Delannoy numbers A001850.

Original entry on oeis.org

1, 4, 20, 112, 664, 4064, 25376, 160640, 1027168, 6618496, 42904960, 279503360, 1828222720, 11999226880, 78984381440, 521218322432, 3447059138048, 22840932997120, 151607254267904, 1007830488424448, 6708862677274624

Offset: 0

Emanuele Munarini, Feb 26 2003

The Hankel transform (see A001906 for definition) of this sequence is A036442: 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
Coefficient of x^n in (1 + 4*x + 2*x^2)^n - N-E. Fahssi, Jan 17 2008
Number of paths from (0,0) to (n,0) using only steps U=(1,1), H=(1,0) and D=(1,-1), U can have 2 colors and H can have 4 colors. - N-E. Fahssi, Jan 27 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Table[SeriesCoefficient[Series[1/Sqrt[1-8x+8x^2], {x, 0, n}], n], {n, 0, 12}]
Table[LegendreP[n, Sqrt[2]] 8^(n/2), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

x='x+O('x^66); Vec(1/sqrt(1-8*x+8*x^2)) \\ Joerg Arndt, May 07 2013

G.f.: 1 / sqrt( 1 - 8*x + 8*x^2 ).
a(n) = Sum_{k=0..n} binomial(n,k) * A001850(k).
E.g.f.: exp(4*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004
Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 13 2012
G.f.: G(0), where G(k)= 1 + 4*x*(1-x)*(4*k+1)/(2*k+1 - 2*x*(1-x)*(2*k+1)*(4*k+3)/(2*x*(1-x)*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013
a(n) = LegendreP_n(sqrt(2))*8^(n/2). - Vladimir Reshetnikov, Nov 01 2015

A217628 a(n) = 3^((n-1)*(n+2)/2).

Original entry on oeis.org

1, 9, 243, 19683, 4782969, 3486784401, 7625597484987, 50031545098999707, 984770902183611232881, 58149737003040059690390169, 10301051460877537453973547267843, 5474401089420219382077155933569751763, 8727963568087712425891397479476727340041449

Offset: 1

Vincenzo Librandi, Oct 24 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..60

Cf. A036442.

I:=[1]; [n le 1 select I[n] else Self(n-1)*3^n: n in [1..20]]

Mathematica
```
Table[3^((n-1) * (n+2)/2), {n, 1, 30}]
```

A217628[n]:=3^((n-1)*(n+2)/2)$
makelist(A217628[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

a(1) = 1, a(n) = a(n-1) * 3^n.

G.f.: G(0)/(2*x^3) - 1/(3*x)- 1/(3*x^2)- 1/(2*x^3), where G(k)= 1 + 3^(k-1)*x/(1 - 1/(1 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 28 2013

A344536 Inverse permutation to A344534.

Original entry on oeis.org

0, 1, 4, 2, 32, 5, 512, 3, 16, 33, 16384, 6, 1048576, 513, 36, 8, 134217728, 17, 34359738368, 34, 516, 16385, 17592186044416, 7, 256, 1048577, 20, 514, 18014398509481984, 37, 36893488147419103232, 9, 16388, 134217729, 544, 18, 151115727451828646838272

Offset: 1

Rémy Sigrist, May 23 2021

This sequence is additive.

This sequence establishes a bijection from A225547 to A261195.

			A344534(42) = 320, so a(320) = 42.

Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A006125, A036442, A064547, A225546, A225547, A261195, A344534, A344537.

a(n) = { my (f=factor(n), v=0); for (k=1, #f~, my (x=primepi(f[k,1])-1, yy=f[k,2], y); while (yy, yy-=2^y=valuation(yy,2); v+=2^(x + (x+y)*
(x+y+1)/2))); v }

a(prime(n)) = A036442(n).
a(2^2^n) = A006125(n+1) for any n >= 0.
A000120(a(n)) = A064547(n).
a(A225546(n)) = A344537(n).
a(n) = A344537(n) iff n belongs to A225547.

A344537 Inverse permutation to A344535.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 64, 5, 16, 9, 1024, 6, 32768, 65, 10, 32, 2097152, 17, 268435456, 12, 66, 1025, 68719476736, 7, 128, 32769, 18, 68, 35184372088832, 11, 36028797018963968, 33, 1026, 2097153, 72, 20, 73786976294838206464, 268435457, 32770, 13

Offset: 1

Rémy Sigrist, May 23 2021

This sequence is additive.

This sequence establishes a bijection from A225547 to A261195.

			A344535(42) = 240, so a(240) = 42.

Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A006125, A036442, A064547, A225546, A225547, A261195, A344535, A344536.

a(n) = { my (f=factor(n), v=0); for (k=1, #f~, my (x=primepi(f[k, 1])-1, yy=f[k, 2], y); while (yy, yy-=2^y=valuation(yy, 2); v+=2^(y + (x+y)*(x+y+1)/2))); v }

a(prime(n)) = A006125(n+1) for any n >= 0.
a(2^2^n) = A036442(n).
A000120(a(n)) = A064547(n).
a(A225546(n)) = A344536(n).
a(n) = A344536(n) iff n belongs to A225547.

A352782 The binary expansion of a(n) encodes the runs of consecutive 1's in the binary expansion of n (see Comments section for precise definition).

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 32, 64, 5, 6, 12, 128, 9, 256, 512, 1024, 17, 10, 20, 24, 7, 48, 96, 2048, 33, 18, 36, 4096, 65, 8192, 16384, 32768, 129, 34, 68, 40, 11, 80, 160, 192, 13, 14, 28, 384, 25, 768, 1536, 65536, 257, 66, 132, 72, 19, 144, 288, 131072, 513

Offset: 0

Rémy Sigrist, Apr 02 2022

For any nonnegative integer n:
- the binary expansion of n can be uniquely expressed as the concatenation of k = A069010(n) positive terms of A023758 separated by 0's:
      n = A023758(m_k+1) | 0 | A023758(m_{k-1}+1) | 0 | ... | 0 | A023758(m_1+1)
            (where | denotes binary concatenation)
- a(n) = ( Sum_{i = 1..k} 2^Sum_{j = 1..i} m_j ) / 2.
This sequence is a permutation of the nonnegative integers, with inverse A352783.

			For n = 89:
- the binary expansion of 89 is "1011001",
- "1011001" = "1" | 0 | "110" | 0 | "1"
            = A023758(1+1) | 0 | A023758(5+1) | 0 | A023758(1+1)
- so 2*a(89) = 2^(1+5+1) + 2^(5+1) + 2^1 = 194,
- and a(89) = 97.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Colored logarithmic scatterplot of the first 2^20 terms (where the color is function of A069010(n))
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A006125, A023758, A036442, A069010, A131865, A352783 (inverse).

a(n) = { my (v=0, s=-1, z, o, i); while (n, n\=2^z=valuation(n,2); n\=2^o=valuation(n+1,2); n\=2; i=(o+z)*(o+z-1)/2 + o; v+=2^s+=i); v }

a(4*n+1) = 2*a(n)+1.
A000120(a(n)) = A069010(n).
a(A023758(k+1)) = 2^k for any k >= 0.
a(2^k) = A006125(k+1) for any k >= 0.
a(2^k-1) = A036442(k+1) for any k >= 0.
a(n) = n iff n = 0 or n belongs to A131865 or n/2 belongs to A131865.

A128131 a(n) = 2^A000096(n)*A128130(n).

Original entry on oeis.org

1, -4, 0, 0, -16384, 1048576, 0, 0, 17592186044416, -18014398509481984, 0, 0, -1237940039285380274899124224, 20282409603651670423947251286016, 0, 0, 5708990770823839524233143877797980545530986496

Offset: 0

Paul Barry, Feb 15 2007

Cf. A036442, A127945.

A128131 := proc(n)
    2^(n*(n+3)/2)*A128130(n) ;
end proc: # R. J. Mathar, Feb 24 2015

a(n) = 2^n * 2^C(n+1,2) * A128130(n);

Incorrect NAME removed. - R. J. Mathar, Feb 24 2015

cp's OEIS Frontend

A344534 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+A002262(e_k))^2^A025581(e_k) (where prime(k) denotes the k-th prime number).

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A344535 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+A025581(e_k))^2^A002262(e_k) (where prime(k) denotes the k-th prime number).

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A080609 Binomial transform of central Delannoy numbers A001850.

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Mathematica

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A217628 a(n) = 3^((n-1)*(n+2)/2).

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Magma

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A344536 Inverse permutation to A344534.

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A344537 Inverse permutation to A344535.

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A352782 The binary expansion of a(n) encodes the runs of consecutive 1's in the binary expansion of n (see Comments section for precise definition).

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A128131 a(n) = 2^A000096(n)*A128130(n).

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