A130793 -id:A130793 - cp's OEIS frontend

A029747 Numbers of the form 2^k times 1, 3 or 5.

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40960, 49152, 65536, 81920, 98304, 131072, 163840, 196608

Offset: 1

N. J. A. Sloane

Fixed points of the Doudna sequence: A005940(a(n)) = A005941(a(n)) = a(n). - Reinhard Zumkeller, Aug 23 2006
Subsequence of A103969. - R. J. Mathar, Mar 06 2010
Question: Is there a simple proof that A005940(c) = c would never allow an odd composite c as a solution? See also my comments in A163511 and in A335431 concerning similar problems, also A364551 and A364576. - Antti Karttunen, Jul 28 & Aug 11 2023

			128 = 2^7 * 1 is in the sequence as well as 160 = 2^5 * 5. - _David A. Corneth_, Sep 18 2020

David A. Corneth, Table of n, a(n) for n = 1..9963 (terms <= 10^1000)
Index entries for linear recurrences with constant coefficients, signature (0,0,2).

Cf. A122132, A000079, A007283, A020714, A130793.
Cf. A005940, A005941, A163511, A335431, A364499, A364551, A364576.
Subsequence of the following sequences: A103969, A253789, A364541, A364542, A364544, A364546, A364548, A364550, A364560, A364565.
Even terms form a subsequence of A320674.

m = 200000; Select[Union @ Flatten @ Outer[Times, {1, 3, 5}, 2^Range[0, Floor[Log2[m]]]], # < m &] (* Amiram Eldar, Oct 15 2020 *)

is(n) = n>>valuation(n, 2) <= 5 \\ David A. Corneth, Sep 18 2020

def A029747(n):
    if n<3: return n
    a, b = divmod(n,3)
    return 1<Chai Wah Wu, Apr 02 2025

a(n) = if n < 6 then n else 2*a(n-3). - Reinhard Zumkeller, Aug 23 2006
G.f.: (1+x+x^2)^2/(1-2*x^3). - R. J. Mathar, Mar 06 2010
Sum_{n>=1} 1/a(n) = 46/15. - Amiram Eldar, Oct 15 2020

Edited by David A. Corneth and Peter Munn, Sep 18 2020

A176907 Decimal expansion of (9+sqrt(145))/16.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 28 2010

Continued fraction expansion of (9+sqrt(145))/16 is A130793.

			(9+sqrt(145))/16 = 1.31509966117451846750...

Cf. A176910 (decimal expansion of sqrt(145)), A130793 (repeat 1, 3, 5).

A271390 a(n) = (2n + 1)^(2floor((n-1)/2) + 1).

Original entry on oeis.org

1, 3, 5, 343, 729, 161051, 371293, 170859375, 410338673, 322687697779, 794280046581, 952809757913927, 2384185791015625, 4052555153018976267, 10260628712958602189, 23465261991844685929951, 59938945498865420543457, 177482997121587371826171875, 456487940826035155404146917

Offset: 0

Ilya Gutkovskiy, Apr 06 2016

All members are odd, therefore:
........................
|   k   |  a(n) mod k  |
|.......|..............|
|  n+1  |  A001477(n)  |
| 2*n+2 |  A005408(n)  |
|   2   |  A000012(n)  |
|   3   |  A080425(n+2)|
|   4   |  A010684(n)  |
|   6   |  A130793(n)  |
........................
Final digit of (2*n + 1)^(2*floor((n-1)/2) + 1) gives periodic sequence -> period 20: repeat [1,3,5,3,9,1,3,5,3,9,1,7,5,7,9,1,7,5,7,9], defined by the recurrence relation b(n) = b(n-2) - b(n-4) + b(n+5) + b(n+6) - b(n-7) - b(n-8) + b(n-9) - b(n-11) + b(n-13).

			a(0) =  1;
a(1) =  3^1 = 3;
a(2) =  5^1 = 5;
a(3) =  7^3 = 343;
a(4) =  9^3 = 729;
a(5) = 11^5 = 161051;
a(6) = 13^5 = 371293;
a(7) = 15^7 = 170859375;
a(8) = 17^7 = 410338673;
...
a(10000) = 1.644...*10^43006;
...
a(100000) = 8.235...*10^530097, etc.
This sequence can be represented as a binary tree:
                                    1
                 ................../ \..................
                3^1                                   5^1
     7^3......../ \......9^3                11^5....../ \.......13^5
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
15^7    17^7        19^9    21^9        23^11   25^11       27^13   29^13

Ilya Gutkovskiy, Table of n, a(n) for n = 0..75

Cf. A005408, A092503, A109613.

A271390:=n->(2*n + 1)^(n - 1/2 - (-1)^n/2): seq(A271390(n), n=0..30); # Wesley Ivan Hurt, Apr 10 2016

Table[(2 n + 1)^(2 Floor[(n - 1)/2] + 1), {n, 0, 18}]
Table[(2 n + 1)^(n - 1 + (1 + (-1)^(n - 1))/2), {n, 0, 18}]

a(n) = (2*n + 1)^(2*((n-1)\2) + 1); \\ Altug Alkan, Apr 06 2016

for n in range(0,10**3):print((int)((2*n+1)**(2*floor((n-1)/2)+1)))
# Soumil Mandal, Apr 10 2016

a(n) = (2*n + 1)^(n - 1 + (1 + (-1)^(n-1))/2).
a(n) = A005408(n)^A109613(n-1).
a(n) = (2*n + 1)^(n - 1/2 - (-1)^n/2). - Wesley Ivan Hurt, Apr 10 2016

cp's OEIS Frontend

A029747 Numbers of the form 2^k times 1, 3 or 5.

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A176907 Decimal expansion of (9+sqrt(145))/16.

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A271390 a(n) = (2n + 1)^(2floor((n-1)/2) + 1).

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cp's OEIS Frontend

A029747 Numbers of the form 2^k times 1, 3 or 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A176907 Decimal expansion of (9+sqrt(145))/16.

Views

Author

Keywords

Comments

Examples

Crossrefs

A271390 a(n) = (2*n + 1)^(2*floor((n-1)/2) + 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A271390 a(n) = (2n + 1)^(2floor((n-1)/2) + 1).