A005010 -id:A005010 - cp's OEIS frontend

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 1

Enoch Haga, Jan 24 2004

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing digit "1".

The monotonic increasing value of successive product of digits strongly suggests that in successive n the digit 1 must be present.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
Carlos Rivera, Puzzle 251, Pointer primes, The Prime Puzzles and Problems Connection.
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A089823, A091628, A091630, A091631, A091632.
Similar to A003945, A042950, A058764, A087009, A081808.

[3*2^n : n in [1..40]]; // Wesley Ivan Hurt, Jul 17 2020

3*2^Range[1, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

[3*2^n for n in range(1,51)] # G. C. Greubel, Jan 05 2023

a(n) = 3 * 2^n = product of digits of A091628(n).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 6*2^(n-1).
a(n) = 2*a(n-1), with a(1) = 6.
G.f.: 6*x/(1-2*x). (End)
E.g.f.: 3*(exp(2*x) - 1). - G. C. Greubel, Jan 05 2023

Edited and extended by Ray Chandler, Feb 07 2004

A248461 T(n,k)=Number of length n+2 0..k arrays with no three consecutive terms having the sum of any two elements equal to twice the third.

Original entry on oeis.org

6, 18, 10, 48, 36, 16, 96, 148, 72, 26, 174, 380, 460, 144, 42, 282, 862, 1512, 1436, 288, 68, 432, 1652, 4272, 6040, 4488, 576, 110, 624, 2956, 9684, 21182, 24160, 14040, 1152, 178, 870, 4860, 20236, 56782, 105026, 96736, 43940, 2304, 288, 1170, 7642, 37868

Offset: 1

R. H. Hardin, Oct 06 2014

Table starts
...6...18......48.......96.......174........282.........432.........624
..10...36.....148......380.......862.......1652........2956........4860
..16...72.....460.....1512......4272.......9684.......20236.......37868
..26..144....1436.....6040.....21182......56782......138534......295078
..42..288....4488....24160....105026.....332940......948412.....2299356
..68..576...14040....96736....520788....1952254.....6493036....17917712
.110.1152...43940...387488...2582406...11447368....44452660...139623544
.178.2304..137532..1552448..12805334...67123652...304332258..1088015294
.288.4608..430508..6220480..63497776..393591402..2083523194..8478351478
.466.9216.1347652.24926080.314866606.2307892826.14264241960.66067495706

			Some solutions for n=5 k=4
..3....4....4....0....1....4....3....1....0....0....1....1....1....1....3....4
..4....4....0....1....3....1....2....1....2....3....1....0....3....3....0....1
..4....1....4....1....0....2....3....4....3....1....0....1....4....4....3....1
..3....4....4....0....1....1....3....1....2....0....1....0....0....0....3....4
..0....0....1....4....1....2....2....4....3....1....4....3....0....0....4....1
..3....4....2....0....2....1....3....1....3....4....4....4....3....1....1....4
..3....3....4....0....2....1....2....2....4....2....0....4....1....3....1....4

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A006355(n+4)

Column 2 is A005010

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1)
k=3: a(n) = 2*a(n-1) +3*a(n-2) +4*a(n-3) -3*a(n-4) -12*a(n-5) -4*a(n-6)
k=4: a(n) = 3*a(n-1) +5*a(n-2) +2*a(n-3) -16*a(n-4) -28*a(n-5) -8*a(n-6)
k=5: [order 12]
k=6: [order 16]
k=7: [order 22]
Empirical for row n:
n=1: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5); also a cubic polynomial plus a constant quasipolynomial with period 2
n=2: a(n) = 2*a(n-1) -a(n-3) -2*a(n-5) +2*a(n-6) +a(n-8) -2*a(n-10) +a(n-11); also a quartic polynomial plus a linear quasipolynomial with period 12
n=3: [order 27; also a degree 5 polynomial plus a quadratic quasipolynomial with period 840]
n=4: [order 61]

A250769 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing absolute value of x(i,j)-x(i-1,j) in the j direction.

Original entry on oeis.org

9, 18, 18, 35, 34, 36, 68, 62, 66, 72, 133, 114, 114, 130, 144, 262, 214, 196, 216, 258, 288, 519, 410, 344, 350, 418, 514, 576, 1032, 798, 622, 572, 648, 820, 1026, 1152, 2057, 1570, 1158, 962, 996, 1234, 1622, 2050, 2304, 4106, 3110, 2208, 1680, 1558, 1812

Offset: 1

R. H. Hardin, Nov 27 2014

Table starts
....9...18....35....68...133...262...519..1032..2057..4106...8203..16396..32781
...18...34....62...114...214...410...798..1570..3110..6186..12334..24626..49206
...36...66...114...196...344...622..1158..2208..4284..8410..16634..33052..65856
...72..130...216...350...572...962..1680..3046..5700.10922..21272..41870..82956
..144..258...418...648...996..1558..2526..4284..7600.14010..26586..51472.100956
..288..514...820..1234..1812..2666..4020..6322.10468.18250..33252..62642.120756
..576.1026..1622..2396..3412..4798..6810..9960.15272.24794..42622..76948.144156
.1152.2050..3224..4710..6580..8978.12192.16798.23948.35946..57400..97526.174756
.2304.4098..6426..9328.12884.17254.22758.30036.40368.56314..82994.130648.219756
.4608.8194.12828.18554.25460.33722.43692.56074.72276.95114.130220.188858.293556

			Some solutions for n=4 k=4
..1..1..1..1..0....1..0..0..0..0....1..0..1..1..0....1..1..0..1..1
..1..1..1..1..0....1..1..1..1..1....1..0..1..1..0....1..1..0..1..1
..1..1..1..1..0....0..0..0..0..0....1..0..1..1..1....1..1..0..1..1
..1..1..1..1..0....0..0..0..0..1....1..0..1..1..1....1..1..0..1..1
..0..0..0..0..1....0..0..0..0..1....1..0..1..1..1....1..1..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..611

Column 1 is A005010(n-1)
Column 2 is A052548(n+3)
Row 1 is A083706(n+1)

Empirical for column k: (k+2)^2*2^(n-1) plus a linear polynomial in n
k=1: a(n) = 2*a(n-1); a(n) = 9*2^(n-1)
k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 16*2^(n-1) + 2
k=3: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 25*2^(n-1) + 2*n + 8
k=4: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 36*2^(n-1) + 10*n + 22
k=5: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 49*2^(n-1) + 32*n + 52
k=6: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 64*2^(n-1) + 84*n + 114
k=7: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 81*2^(n-1) + 198*n + 240
k=8: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 100*2^(n-1) + 438*n + 494
k=9: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 121*2^(n-1) + 932*n + 1004
Empirical for row n: (4*n+4)*2^(k-1) plus a quadratic polynomial in k
n=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 8*2^(n-1) + n
n=2: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 12*2^(n-1) + 4*n + 2
n=3: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 16*2^(n-1) + n^2 + 11*n + 8
n=4: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 20*2^(n-1) + 4*n^2 + 26*n + 22
n=5: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 24*2^(n-1) + 11*n^2 + 57*n + 52
n=6: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 28*2^(n-1) + 26*n^2 + 120*n + 114
n=7: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 32*2^(n-1) + 57*n^2 + 247*n + 240
n=8: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 36*2^(n-1) + 120*n^2 + 502*n + 494
n=9: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 40*2^(n-1) + 247*n^2 + 1013*n + 1004

A090570 Numbers that are congruent to {0, 1} mod 9.

Original entry on oeis.org

0, 1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99, 100, 108, 109, 117, 118, 126, 127, 135, 136, 144, 145, 153, 154, 162, 163, 171, 172, 180, 181, 189, 190, 198, 199, 207, 208, 216, 217, 225, 226

Offset: 1

Giovanni Teofilatto, Feb 25 2004

			13 is 1101 in base 2, so a(13+1) = a(14) = 36*1 + 18*1 + 9*0 + 1*1 = 36+18+1 = 55. - _Philippe Deléham_, Oct 17 2011

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

Union of A008591 and A017173. - Reinhard Zumkeller, Oct 10 2008
Cf. A010888, A145389.
Cf. A005010, A030308.

forstep(n=0,200,[1,8],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

A145389(a(n)) = A010888(a(n)). - Reinhard Zumkeller, Oct 10 2008
a(n) = 9*n - a(n-1) - 17 (with a(1)=0). - Vincenzo Librandi, Nov 16 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 9*n/2 - 25/4 - 7*(-1)^n/4.
G.f.: x^2*(1+8*x)/( (1+x)*(1-x)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*A005010(k-1), with A005010(-1)=1. - Philippe Deléham, Oct 17 2011.
E.g.f.: 8 + ((18*x - 25)*exp(x) - 7*exp(-x))/4. - David Lovler, Sep 03 2022

A110287 a(n) = 17*2^n.

Original entry on oeis.org

17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 8704, 17408, 34816, 69632, 139264, 278528, 557056, 1114112, 2228224, 4456448, 8912896, 17825792, 35651584, 71303168, 142606336, 285212672, 570425344, 1140850688, 2281701376, 4563402752, 9126805504, 18253611008

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

17 times powers of 2. - Omar E. Pol, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A003945 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), this sequence (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

Cf. A007283.

[17*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

17*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,17,30] (* Harvey P. Dale, Jul 21 2014 *)

a(n)=17*2^n \\ Charles R Greathouse IV, Oct 07 2015

[17*2^n for n in range(51)] # G. C. Greubel, Jan 05 2023

G.f.: 17/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = 17*A000079(n). - Omar E. Pol, Dec 17 2008
a(n) = 2*a(n-1) (with a(0)=17). - Vincenzo Librandi, Dec 26 2010
a(n) = A173786(n+4, n) for n>3. - Reinhard Zumkeller, Feb 28 2010
E.g.f.: 17*exp(2*x). - G. C. Greubel, Jan 05 2023

Edited by Omar E. Pol, Dec 16 2008

A116453 Third smallest number with exactly n prime factors.

Original entry on oeis.org

5, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 147456, 294912, 589824, 1179648, 2359296, 4718592, 9437184, 18874368, 37748736, 75497472, 150994944, 301989888, 603979776, 1207959552, 2415919104

Offset: 1

Reinhard Zumkeller, Feb 16 2006

Smallest term in A116451 having exactly n prime factors;
a(n) = A116451(A116454(n)) = A116454(n) + 2 * n + 3;
a(n) = A005010(n-2) = 9*2^(n - 2) for n > 1.

			a(1) =  5 = A000040(3) > A000040(2) =  3 > A000040(1) =  2;
a(2) =  9 = A001358(3) > A001358(2) =  6 > A001358(1) =  4;
a(3) = 18 = A014612(3) > A014612(2) = 12 > A014612(1) =  8;
a(4) = 36 = A014613(3) > A014613(2) = 24 > A014613(1) = 16.

Index entries for linear recurrences with constant coefficients, signature (2).

def A116453(n): return 9<1 else 5 # Chai Wah Wu, Aug 23 2024

A279634 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 3/2.

Original entry on oeis.org

1, -3, 5, -9, 18, -36, 72, -144, 288, -576, 1152, -2304, 4608, -9216, 18432, -36864, 73728, -147456, 294912, -589824, 1179648, -2359296, 4718592, -9437184, 18874368, -37748736, 75497472, -150994944, 301989888, -603979776, 1207959552, -2415919104, 4831838208

Offset: 0

Clark Kimberling, Dec 18 2016

After first 3 terms, agrees with A005010 except for signs; in particular 9 divides a(n) for n >= 3.
Suppose r = c/d is a rational number and (a(n)) is the coefficient series for 1/([r] + [2r]x + [3r]x^2 + ...). Let (s(k)) be the increasing sequence of indices n(k) for which a(n(k)) > = 0. In the table below, "yes" indicates that a check of the first 1000 terms indicates that (n(k)) is (eventually) periodic. Column 1 gives selected values of r, and column 2 gives the corresponding coefficient series.
3/2    A279634     yes
4/3    A279675     no
5/3    A279676     no
5/4    A279677     yes
7/4    A279678     yes
6/5    A279778     no
7/5    A279779     no
8/5    A279780     yes
9/5    A279781     no

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A005010.

z = 50; f[x_] := f[x] = Sum[Floor[(3/2)*(k + 1)] x^k, {k, 0, z}]; f[x]
CoefficientList[Series[1/f[x], {x, 0, z}], x]
LinearRecurrence[{-2},{1,-3,5,-9},40] (* Harvey P. Dale, Jul 28 2023 *)

G.f.: 1/(1 + 3x + 4x^2 + 6x^3 + ...).
G.f.: (1 - x) (1 - x^2)/(1 + 2x).
E.g.f.: - (1/8) - (3/4)*x + (1/4)*x^2 + (9/8)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 16 2021

A110288 a(n) = 19*2^n.

Original entry on oeis.org

19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 19456, 38912, 77824, 155648, 311296, 622592, 1245184, 2490368, 4980736, 9961472, 19922944, 39845888, 79691776, 159383552, 318767104, 637534208, 1275068416, 2550136832, 5100273664, 10200547328, 20401094656

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

19 times powers of 2. - Omar E. Pol, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), this sequence (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

[19*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

19*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,19,30] (* Harvey P. Dale, May 11 2018 *)

[19*2^n for n in range(41)] # G. C. Greubel, Jan 04 2023

G.f.: 19/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = A000079(n)*19. - Omar E. Pol, Dec 17 2008
E.g.f.: 19*exp(2*x). - G. C. Greubel, Jan 04 2023

Edited by Omar E. Pol, Dec 16 2008

A327539 Starting from n: as long as the decimal representation starts with a positive even number, divide the largest such prefix by 2; a(n) corresponds to the final value.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 11, 11, 13, 3, 15, 13, 17, 7, 19, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 11, 11, 13, 11, 15, 13, 17, 3, 19, 15, 51, 13, 53, 17, 55, 7, 57, 19, 59, 15, 31, 31, 33, 1, 35, 33, 37, 17, 39, 35, 71

Offset: 0

Rémy Sigrist, Nov 29 2019

For n > 0, as long as we have a number whose decimal representation is the concatenation of a positive even number, say u, and a possibly empty string of odd digits, say v, we replace this number with the concatenation of u/2 and v; eventually only odd digits remain.

			For n = 10000:
- 10000 gives 10000/2 = 5000,
- 5000 gives 5000/2 = 2500,
- 2500 gives 2500/2 = 1250,
- 1250 gives 125/2 = 625,
- 625 gives 62/2 followed by 5 = 315,
- 315 has only odd digits, so a(10000) = 315.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Logarithmic scatterplot of the first 1000000 terms
Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

See A329249, A329424 and A329428 for similar sequences.

Cf. A007283, A014261, A020714, A005009, A005010, A215145.

Array[FixedPoint[If[AllTrue[#, OddQ], FromDigits@ #, FromDigits@ Flatten@ Join[IntegerDigitsFromDigits[First[#]]/2, Last[#]] &@ TakeDrop[#, Position[#, ?EvenQ][[-1, -1]] ] ] &@ IntegerDigits[#] &, #] &, 71] (* _Michael De Vlieger, Dec 01 2019 *)

a(n) = if (n==0, 0, n%2==0, a(n/2), 10*a(n\10)+(n%10))

a(n) <= n with equality iff n = 0 or n belongs to A014261.
a(2*n) = a(n).
a(10*k + v) = 10*a(k) + v for any k >= 0 and v in {1, 3, 5, 7, 9}.
a(n) = 1 iff n is a power of 2.
a(n) = 3 iff n belongs to A007283.
a(n) = 5 iff n belongs to A020714.
a(n) = 7 iff n belongs to A005009.
a(n) = 9 iff n belongs to A005010.
a(n) = a(n+1) iff n belongs to A215145.

A140683 a(n) = 3(-1)^(n+1)2^n - 1.

Original entry on oeis.org

-4, 5, -13, 23, -49, 95, -193, 383, -769, 1535, -3073, 6143, -12289, 24575, -49153, 98303, -196609, 393215, -786433, 1572863, -3145729, 6291455, -12582913, 25165823, -50331649, 100663295, -201326593, 402653183, -805306369, 1610612735, -3221225473

Offset: 0

Paul Curtz, Jul 11 2008

sign

Alternated reading of negative of A140660 and A140529.
The binomial transform yields -4 followed by the negative of A140657.
The inverse binomial transform yields essentially a signed version of A000244. - R. J. Mathar, Aug 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 2).

[3*(-1)^(n+1)*2^n-1: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

Table[3(-1)^(n+1)2^n-1,{n,0,40}] (* or *) LinearRecurrence[{-1,2},{-4,5},40] (* Harvey P. Dale, May 26 2011 *)

a(2n) = -A140660(n). a(2n+1) = A140529(n).
a(n+1) - a(n) = (-1)^n*A005010(n). a(2n) + a(2n+1) = A096045(n).
a(n) = A140590(n+1) - 2*A140590(n).
O.g.f: (4-x)/((x-1)(2x+1)). - R. J. Mathar, Aug 02 2008
a(n) = -a(n-1) + 2*a(n-2); a(0)=-4, a(1)=5. - Harvey P. Dale, May 26 2011

Edited and extended by R. J. Mathar, Aug 02 2008

cp's OEIS Frontend

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

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A248461 T(n,k)=Number of length n+2 0..k arrays with no three consecutive terms having the sum of any two elements equal to twice the third.

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A250769 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing absolute value of x(i,j)-x(i-1,j) in the j direction.

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A090570 Numbers that are congruent to {0, 1} mod 9.

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PARI

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A110287 a(n) = 17*2^n.

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A116453 Third smallest number with exactly n prime factors.

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A279634 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 3/2.

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A110288 a(n) = 19*2^n.

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A140683 a(n) = 3(-1)^(n+1)2^n - 1.