A110286 -id:A110286 - cp's OEIS frontend

A005009 a(n) = 7*2^n.

7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, 14336, 28672, 57344, 114688, 229376, 458752, 917504, 1835008, 3670016, 7340032, 14680064, 29360128, 58720256, 117440512, 234881024, 469762048, 939524096, 1879048192, 3758096384

Offset: 0

N. J. A. Sloane

The first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini, Sep 07 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
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), this sequence (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).
Row sums of (6, 1)-Pascal triangle A093563 and of (1, 6)-Pascal triangle A096956, n>=1.
Cf. A000120, A014311, A118416, A173787, A212191.

a005009 = (* 7) . (2 ^) -- Reinhard Zumkeller, May 03 2012

[7*2^n:n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

7*2^Range[0,50] (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)
NestList[2#&,7,30] (* Harvey P. Dale, Aug 10 2024 *)

a(n)=7<Charles R Greathouse IV, Dec 22 2011

[7*2^n for n in range(51)] # G. C. Greubel, Jan 05 2023

G.f.: 7/(1-2*x).
a(n) = A118416(n+1,4) for n > 3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 2*a(n-1), for n > 0, with a(0)=7 . - Philippe Deléham, Nov 23 2008
a(n) = 7 * A000079(n). - Omar E. Pol, Dec 16 2008
a(n) = A173787(n+3,n). - Reinhard Zumkeller, Feb 28 2010
Intersection of A014311 and A212191: all terms and their squares are the sum of exactly three distinct powers of 2, A000120(a(n)) = A000120(a(n)^2) = 3. - Reinhard Zumkeller, May 03 2012
G.f.: 2/x/G(0) - 1/x + 9, where G(k)= 1 + 1/(1 - x*(7*k+2)/(x*(7*k+9) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
E.g.f.: 7*exp(2*x). - Stefano Spezia, May 15 2021

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Reinhard Zumkeller, Feb 28 2010

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

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

Original entry on oeis.org

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

A195744 a(n) = 15*2^(n+1) + 1.

Original entry on oeis.org

31, 61, 121, 241, 481, 961, 1921, 3841, 7681, 15361, 30721, 61441, 122881, 245761, 491521, 983041, 1966081, 3932161, 7864321, 15728641, 31457281, 62914561, 125829121, 251658241, 503316481, 1006632961, 2013265921, 4026531841, 8053063681, 16106127361

Offset: 0

Brad Clardy, Sep 23 2011

Binary numbers of form 1111(0^n)1 where n is the index and number of 0's.
Base 10 numbers of this sequence always end in 1.
An Engel expansion of 1/15 to the base 2 as defined in A181565, with the associated series expansion 1/15 = 2/31 + 2^2/(31*61) + 2^3/(31*61*121) + 2^4/(31*61*121*241) + ... . - Peter Bala, Oct 29 2013
The only squares in this sequence are 121 = 11^2 and 961 = 31^2. - Antti Karttunen, Sep 24 2023

			First few terms in binary are 11111, 111101, 1111001, 11110001, 111100001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052996, A164094.
Cf. A020737, A083575, A083683, A083686, A083705, A168596, A181565.
Cf. A110286, A128470.
Cf. A005940, A030514.

[15*2^(n+1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 24 2011

15*2^Range[50] + 1 (* Paolo Xausa, Apr 02 2024 *)

a(n)=30*2^n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A052996(n+3) + A164094(n+2).
From Bruno Berselli, Sep 23 2011: (Start)
G.f.: (31-32*x)/(1-3*x+2*x^2).
a(n) = 2*a(n-1)-1.
a(n) = A110286(n+1)+1 = A128470(2^n). (End)
E.g.f.: exp(x)*(1 + 30*exp(x)). - Stefano Spezia, Oct 08 2022
For n >= 0, A005940(a(n)) = A030514(2+n). - Antti Karttunen, Sep 24 2023

Corrected by Arkadiusz Wesolowski, Sep 23 2011

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

A051916 The Greek sequence: 2^a * 3^b * 5^c where a = 0,1,2,3,..., b,c in {0,1}, excluding the terms 1,2; that is: (a,b,c) != (0,0,0), (1,0,0).

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 128, 160, 192, 240, 256, 320, 384, 480, 512, 640, 768, 960, 1024, 1280, 1536, 1920, 2048, 2560, 3072, 3840, 4096, 5120, 6144, 7680, 8192, 10240, 12288, 15360, 16384, 20480

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999

From Reinhard Zumkeller, Mar 19 2010: (Start)
Union of A007283, A020707, A020714, and A110286.
Intersection of A051037 and A003401 apart from terms 1 and 2. (End)

George E. Martin, Geometric Constructions, New York: Springer, 1997, p. 140.

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

Cf. A003401, A007283, A020707, A020714, A051037, A110286.

CoefficientList[Series[x(3x^7+2x^6+2x^5+2x^4+6x^3+5x^2+4x+3)/(1-2x^4),{x,0,60}],x] (* Harvey P. Dale, Dec 23 2012 *)

Vec(x*(3*x^7+2*x^6+2*x^5+2*x^4+6*x^3+5*x^2+4*x+3)/(1-2*x^4)+O(x^99)) \\ Charles R Greathouse IV, Oct 12 2012

def A051916(n): return n+2 if n<5 else (15,1,5,3)[m:=n&3]<<(n>>2)+(-2,2,0,1)[m] # Chai Wah Wu, Apr 02 2025

G.f.: x*(3*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 6*x^3 + 5*x^2 + 4*x + 3)/(1 - 2*x^4).
a(n+4) = 2*a(n) for n > 8. - Reinhard Zumkeller, Mar 19 2010
Sum_{n>=1} 1/a(n) = 17/10. - Amiram Eldar, Jan 18 2023

More terms from James Sellers, Dec 18 1999

Offset corrected by Reinhard Zumkeller, Mar 10 2010

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

A196305 a(n) = 15*2^n - 1.

Original entry on oeis.org

14, 29, 59, 119, 239, 479, 959, 1919, 3839, 7679, 15359, 30719, 61439, 122879, 245759, 491519, 983039, 1966079, 3932159, 7864319, 15728639, 31457279, 62914559, 125829119, 251658239, 503316479, 1006632959, 2013265919, 4026531839, 8053063679, 16106127359, 32212254719, 64424509439

Offset: 0

Brad Clardy, Oct 07 2011

Primes of this sequence are in A196940.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A033484, A083705, A110286 (first differences), A196940.

Magma
```
[15*2^n -1 : n in [0..50]];
```

a(n)=15<Charles R Greathouse IV, Oct 08 2011

a(n) = 15*2^n - 1.
a(n) = A033484(n+1) + A083705(n).
From Philippe Deléham, Feb 17 2014: (Start)
a(n) = 2*a(n-1) + 1.
a(n) = 3*a(n-2) - 2*a(n-2).
a(n) = A110286(n) - 1. (End)
From Elmo R. Oliveira, Sep 14 2024: (Start)
G.f.: (14 - 13*x)/((1 - x)*(1 - 2*x)).
E.g.f.: exp(x)*(15*exp(x) - 1). (End)

A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

Original entry on oeis.org

60, 84, 120, 132, 140, 150, 156, 168, 204, 220, 228, 240, 260, 264, 270, 276, 280, 294, 300, 308, 312, 315, 336, 340, 348, 364, 372, 378, 380, 408, 420, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 528, 532, 540, 552, 560, 564, 572, 580, 585, 588

Offset: 1

Gus Wiseman, Dec 25 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The terms together with their Heinz partitions begin (A-E = 10-14):
     60: (3211)      276: (9211)      420: (43211)
     84: (4211)      280: (43111)     440: (53111)
    120: (32111)     294: (4421)      444: (C211)
    132: (5211)      300: (33211)     456: (82111)
    140: (4311)      308: (5411)      460: (9311)
    150: (3321)      312: (62111)     476: (7411)
    156: (6211)      315: (4322)      480: (3211111)
    168: (42111)     336: (421111)    490: (4431)
    204: (7211)      340: (7311)      492: (D211)
    220: (5311)      348: (A211)      495: (5322)
    228: (8211)      364: (6411)      516: (E211)
    240: (321111)    372: (B211)      520: (63111)
    260: (6311)      378: (42221)     528: (521111)
    264: (52111)     380: (8311)      532: (8411)
    270: (32221)     408: (72111)     540: (322211)

Including all nonsquarefree numbers gives A013929, complement A005117.
Subsets include A088860 and A110286.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A336568, counted by A347548.
A version for compositions allowing strict is A349057, counted by A349053.
Allowing strict partitions gives A349794, counted by A349061.
These partitions are counted by A349796.
The complement in nonsquarefree partitions is A350137, counted by A349795.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A096441 = weakly alternating 0-appended partitions.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052/A129852/A129853 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, strong A349801.
A349798 = weakly but not strongly alternating perms of prime indices.
Cf. A000111, A047967, A333213, A335448, A344615, A344653, A345173, A349054, A349059, A349797, A349799.

Select[Range[300],!SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

Complement of A005117 in A349794.

A253145 Triangular numbers (A000217) omitting the term 1.

Original entry on oeis.org

0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275

Offset: 0

Paul Curtz, Mar 23 2015

The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
0,      3,    6,  10,   15,  21,  28, 36, ...
-3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378
3,     12,   24,  40,   60,  84, ...         essentially  A046092
-9,   -24,  -48, -80, -120, ...              essentially -A033996
15,    48,   96, 160, ...
-33,  -96, -192, ...
63,   192, ...
-129, ...
etc.
First column: (-1)^n*A062510(n).
The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n).
An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
The successive differences of b(n) are
0,   0,  0, 3, 6, 10, 15, 21, ...
0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n)
0,   3,  0, 1, 1,  1,  1,  1, ...
3,  -3,  1, 0, 0,  0,  0,  0, ...
-6,  4, -1, 0, 0,  0,  0,  0, ...
10, -5,  1, 0, 0,  0,  0,  0, ...
-15, 6, -1, 0, 0,  0,  0,  0, ...
21, -7,  1, 0, 0,  0,  0,  0, ...
The inverse binomial transform (first column) is the signed sequence. This is general.
Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.

Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018

Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *)

a(n)=if(n,(n+1)*(n+2)/2,0) \\ Charles R Greathouse IV, Mar 23 2015

Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
a(n) = (n+1)*(n+2)/2 for n > 0. - Charles R Greathouse IV, Mar 23 2015
a(n+1) = 3*A001840(n+1) + A022003(n).
a(n) = A161680(n+2) for n >= 1. - Georg Fischer, Oct 30 2018
From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(3 - 3*x + x^2)/(1 - x)^3.
E.g.f.: exp(x)*(2 + 4*x + x^2)/2 - 1. (End)

cp's OEIS Frontend

A005009 a(n) = 7*2^n.

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A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

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

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A195744 a(n) = 15*2^(n+1) + 1.

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

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A051916 The Greek sequence: 2^a * 3^b * 5^c where a = 0,1,2,3,..., b,c in {0,1}, excluding the terms 1,2; that is: (a,b,c) != (0,0,0), (1,0,0).

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