A164095 -id:A164095 - cp's OEIS frontend

A164096 Partial sums of A164095.

5, 11, 21, 33, 53, 77, 117, 165, 245, 341, 501, 693, 1013, 1397, 2037, 2805, 4085, 5621, 8181, 11253, 16373, 22517, 32757, 45045, 65525, 90101, 131061, 180213, 262133, 360437, 524277, 720885, 1048565, 1441781, 2097141, 2883573, 4194293, 5767157

Offset: 1

Klaus Brockhaus, Aug 10 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 2, -2).

Cf. A164095.

T:=[ n le 2 select n+4 else 2*Self(n-2): n in [1..38] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

a(n) = 2*a(n-2)+11 for n > 2; a(1) = 5, a(2) = 11.
a(n) = (27-5*(-1)^n)*2^(1/4*(2*n-1+(-1)^n))/2-11.
G.f.: x*(5+6*x)/(1-x-2*x^2+2*x^3).

A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 5, 3, 4, 1, 5, 4, 7, 2, 6, 3, 7, 2, 4, 5, 8, 3, 11, 4, 11, 1, 6, 5, 6, 4, 13, 7, 10, 2, 13, 6, 17, 3, 8, 7, 17, 2, 9, 4, 10, 5, 19, 8, 6, 3, 14, 11, 19, 4, 23, 11, 12, 1, 10, 6, 23, 5, 14, 6, 29, 4, 29, 13, 8, 7, 9, 10, 31

Offset: 1

Gus Wiseman, Oct 23 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also the Heinz number of the part-wise half (rounded down) of the partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Each n appears A000005(n) times at odd positions (infinitely many at even). To see this, note that our transformation does not distinguish between A066207 and A066208.

			The prime indices of n = 1501500 are {1,1,2,3,3,3,4,5,6}, so the prime indices of a(n) are {1,1,1,1,2,2,3}; hence we have a(1501500) = 720.
The 6 odd positions of 2124 are: 63, 99, 105, 165, 175, 275, with prime indices:
   63: {2,2,4}
   99: {2,2,5}
  105: {2,3,4}
  165: {2,3,5}
  175: {3,3,4}
  275: {3,3,5}

SiYang Hu, Table of n, a(n) for n = 1..10000

Positions of 1's are A000079.
Positions of 2's are 3 and A164095.
Positions of first appearances are A297002, sorted A066207.
A004526 is floor(n/2), with an extra first zero.
A056239 adds up prime indices, row-sums of A112798.
A109763 lists primes of index floor(n/2).
Cf. A003961, A064988, A064989, A076610, A215366, A248601, A357980.

Table[Times@@(If[#1<=2,1,Prime[Floor[PrimePi[#1]/2]]^#2]&@@@FactorInteger[n]),{n,100}]

Completely multiplicative with a(prime(2k)) = prime(k) and a(prime(2k+1)) = prime(k). Cf. A297002.

a(prime(n)) = A109763(n-1).

A164038 Expansion of (5-19x)/(1-10x+23*x^2).

Original entry on oeis.org

5, 31, 195, 1237, 7885, 50399, 322635, 2067173, 13251125, 84966271, 544886835, 3494644117, 22414043965, 143763624959, 922113238395, 5914569009893, 37937085615845, 243335768930911, 1560804720144675, 10011324516035797

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Binomial transform of A161731 without initial 1. Fifth binomial transform of A164095. Inverse binomial transform of A164110.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A161731, A164095, A164110.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(5+r)^n+(5-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

LinearRecurrence[{10,-23},{5,31},50] (* or *) CoefficientList[Series[(5 - 19*x)/(1 - 10*x + 23*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 08 2017 *)

Vec((5-19*x)/(1-10*x+23*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 31.
G.f.: (5-19*x)/(1-10*x+23*x^2).
a(n) = ((5+3*sqrt(2))*(5+sqrt(2))^n + (5-3*sqrt(2))*(5-sqrt(2))^n)/2.
E.g.f: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(5*x). - G. C. Greubel, Sep 08 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 10 2009

A164110 a(n) = 12a(n-1) - 34a(n-2) for n > 1; a(0) = 5, a(1) = 36.

Original entry on oeis.org

5, 36, 262, 1920, 14132, 104304, 771160, 5707584, 42271568, 313200960, 2321178208, 17205305856, 127543611200, 945542935296, 7010032442752, 51971929512960, 385322051101952, 2856819009782784, 21180878379927040

Offset: 0

Klaus Brockhaus, Aug 10 2009

nonn

Binomial transform of A164038. Sixth binomial transform of A164095.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-34).

Cf. A164038, A164095.

[ n le 2 select 31*n-26 else 12*Self(n-1)-34*Self(n-2): n in [1..19] ];

LinearRecurrence[{12,-34}, {5,36}, 50] (* G. C. Greubel, Sep 11 2017 *)

x='x+O('x^50); Vec((5-24*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Sep 11 2017

a(n) = ((5+3*sqrt(2))*(6+sqrt(2))^n+(5-3*sqrt(2))*(6-sqrt(2))^n)/2.
G.f.: (5-24*x)/(1-12*x+34*x^2).
E.g.f.: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(6*x). - G. C. Greubel, Sep 11 2017

A164037 Expansion of (5-9x)/(1-6x+7*x^2).

Original entry on oeis.org

5, 21, 91, 399, 1757, 7749, 34195, 150927, 666197, 2940693, 12980779, 57299823, 252933485, 1116502149, 4928478499, 21755355951, 96032786213, 423909225621, 1871225850235, 8259990522063, 36461362180733, 160948239429957

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Binomial transform of A161941 without initial 2. Third binomial transform of A164095. Inverse binomial transform of A161731 without initial 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A161941, A164095 (5, 6, 10, 12, 20, 24, ...), A161731.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(3+r)^n+(5-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

CoefficientList[Series[(5-9x)/(1-6x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{5,21},30] (* Harvey P. Dale, Apr 27 2017 *)

Vec((5-9*x)/(1-6*x+7*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 5, a(1) = 21.
G.f.: (5-9*x)/(1-6*x+7*x^2).
a(n) = ((5+3*sqrt(2))*(3+sqrt(2))^n+(5-3*sqrt(2))*(3-sqrt(2))^n)/2.
E.g.f.: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 08 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 10 2009

A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Mar 05 2019

nonn

This sequence has graphical features in common with A286326.

			For n=1:
- a(1) = 0 is suitable.
For n=2:
- a(2) = 0 is suitable.
For n=3:
- a(1) = 0 >= a(2) = 0, so a(3) <> 0,
- a(3) = 1 is suitable.
For n=4:
- a(2) = 0 < a(3) = 1,
- a(4) = 0 is suitable.
For n=5:
- a(3) = 1 >= a(4) = 0, so a(5) <> 0,
- a(1) = 0 < a(3) = 1,
- a(5) = 1 is suitable.
For n=6:
- a(4) = 0 < a(5) = 1,
- a(2) = 0 >= a(4) = 0, so a(6) <> 0,
- a(6) = 1 is suitable.
For n=7:
- a(5) = 1 >= a(6) = 1, so a(7) <> 1,
- a(3) = 1 >= a(5) = 1, so a(7) <> 1,
- a(1) = 0 >= a(4) = 0, so a(7) <> 0,
- a(7) = 2 is suitable.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A306696
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 1..100000 (where the color is function of the least k >= 0 such that (1+a(n))/n >= 2^k/A007583(k))

Cf. A000079, A007583, A164095, A181497, A286326.

Empirically:
- a(n) = 0 iff n is a power of 2 (A000079),
- a(n) = 1 iff n = 3 or belongs to A164095,
- a(2*n) = a(n),
- A181497(n) is the least k such that a(k) = n.

cp's OEIS Frontend

A164096 Partial sums of A164095.

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A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.

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Mathematica

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A164038 Expansion of (5-19*x)/(1-10*x+23*x^2).

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A164110 a(n) = 12*a(n-1) - 34*a(n-2) for n > 1; a(0) = 5, a(1) = 36.

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A164037 Expansion of (5-9*x)/(1-6*x+7*x^2).

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Magma

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PARI

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A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A164038 Expansion of (5-19x)/(1-10x+23*x^2).

A164110 a(n) = 12a(n-1) - 34a(n-2) for n > 1; a(0) = 5, a(1) = 36.

A164037 Expansion of (5-9x)/(1-6x+7*x^2).