A168614 -id:A168614 - cp's OEIS frontend

A245318 Numbers k that divide 2^k + 5.

1, 7, 133, 1517, 11761, 676333, 1484413, 3627557, 10289371, 1449045241, 2433687407, 12309023183, 29013950411, 11701492535299, 223598572318157, 362232879754103

Offset: 1

Derek Orr, Jul 17 2014

No other terms below 10^15. Some large terms: 37367159696063084325121, 1637537600494693555095121, 50692913747901869910332539, 407*(2^407+5)/1125038874668278099 (108 digits). - Max Alekseyev, Sep 22 2016

			2^7 + 5 = 133 is divisible by 7. Thus 7 is a term of this sequence.

OEIS Wiki, 2^n mod n

Cf. A128121, A168614.

Select[Range[10^5], Divisible[2^# + 5, #] &] (* Robert Price, Oct 12 2018 *)

for(n=1,10^9,if(Mod(2,n)^n==Mod(-5,n),print1(n,", ")))

a(10)-a(13) from Lars Blomberg, Nov 05 2014

a(14)-a(16) from Max Alekseyev, Oct 09 2016

A254368 a(n) = 5*2^n + 3^n + 15.

Original entry on oeis.org

21, 28, 44, 82, 176, 418, 1064, 2842, 7856, 22258, 64184, 187402, 551936, 1635298, 4864904, 14512762, 43374416, 129795538, 388731224, 1164882922, 3492027296, 10470838978, 31402031144, 94185121882, 282513422576, 847456381618, 2542201372664, 7626268573642, 22878134632256, 68633061719458, 205896500803784, 617684133702202

Offset: 0

Luciano Ancora, Jan 29 2015

This is the sequence of third terms of "fifth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 1.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A168614, A254369, A254370.

Table[5 2^n + 3^n + 15, {n, 0, 30}] (* Bruno Berselli, Jan 30 2015 *)
LinearRecurrence[{6,-11,6},{21,28,44},40] (* Harvey P. Dale, Apr 26 2022 *)

vector(30, n, n--; 5*2^n + 3^n + 15) \\ Colin Barker, Jan 30 2015

G.f.: -(107*x^2-98*x+21) / ((x-1)*(2*x-1)*(3*x-1)). - Colin Barker, Jan 30 2015

a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3). - Colin Barker, Jan 30 2015

A254369 a(n) = 152^n + 4^n + 53^n + 35.

Original entry on oeis.org

56, 84, 156, 354, 936, 2754, 8736, 29274, 102216, 368274, 1359216, 5110794, 19495896, 75203394, 292596096, 1145977914, 4511183976, 17827536114, 70660511376, 280697078634, 1116961278456, 4450379734434, 17749154257056, 70839585900954, 282887376051336, 1130136853206354, 4516309963145136, 18052528510172874, 72171982026734616

Offset: 0

Luciano Ancora, Jan 29 2015

This is the sequence of fourth terms of "fifth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A168614, A254368, A254370.

Table[15 2^n + 4^n + 5 3^n + 35, {n, 0, 30}] (* Bruno Berselli, Jan 30 2015 *)
LinearRecurrence[{10,-35,50,-24},{56,84,156,354},30] (* Harvey P. Dale, Dec 04 2020 *)

vector(30, n, n--; 15*2^n + 4^n + 5*3^n + 35) \\ Colin Barker, Jan 30 2015

G.f.: -2*(533*x^3-638*x^2+238*x-28) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Jan 30 2015

a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4). - Colin Barker, Jan 30 2015

A254370 a(n) = 352^n + 54^n + 15*3^n + 5^n + 70.

Original entry on oeis.org

126, 210, 450, 1200, 3750, 13080, 49350, 197400, 825750, 3577080, 15930150, 72528600, 336141750, 1580449080, 7518010950, 36102667800, 174710721750, 850780489080, 4164115131750, 20465328135000, 100917328245750, 498984369457080, 2472617583932550

Offset: 0

Luciano Ancora, Jan 30 2015

This is the sequence of fifth terms of "fifth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Cf. A168614, A254368, A254369.

Table[35 2^n + 5 4^n + 15 3^n + 5^n + 70, {n, 0, 30}] (* Bruno Berselli, Jan 30 2015 *)

vector(30, n, n--; 35*2^n + 5*4^n + 15*3^n + 5^n + 70) \\ Colin Barker, Jan 30 2015

G.f.: -6*(1879*x^4-2675*x^3+1335*x^2-280*x+21) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Jan 30 2015

a(n) = 15*a(n-1)-85*a(n-2)+225*a(n-3)-274*a(n-4)+120*a(n-5). - Colin Barker, Jan 30 2015

A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

Original entry on oeis.org

21, 37, 53, 69, 101, 117, 133, 197, 229, 245, 261, 389, 453, 485, 501, 517, 773, 901, 965, 997, 1013, 1029, 1541, 1797, 1925, 1989, 2021, 2037, 2053, 3077, 3589, 3845, 3973, 4037, 4069, 4085, 4101, 6149, 7173, 7685, 7941, 8069, 8133, 8165, 8181, 8197, 12293, 14341, 15365, 15877, 16133

Offset: 1

Brad Clardy, Apr 16 2013

The table has row labels 2^n - 1 and column labels 2^(m+3). The table entry is row*col + 5. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
+5  |   16    32    64   128   256    512   1024 ...
----|-------------------------------------------
1   |   21    37    69   133   261    517   1029
3   |   53   101   197   389   773   1541   3077
7   |  117   229   453   901  1797   3589   7173
15  |  245   485   965  1925  3845   7685  15365
31  |  501   997  1989  3973  7941  15877  31749
63  | 1013  2021  4037  8069 16133  32261  64517
127 | 2037  4069  8133 16261 32517  65029 130053
...
All of these numbers have the following property: let m be a member of A(n); if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then the differences between successive members of B(n) is a repeating series
  of 1,1,1,5 ending with 1,1,1 and the last difference in the pattern m. The total number of 1's and 5's in the pattern is 2^(j+2) - 1, where j is the column index.
As an example, consider A(1), which is 21; the sequence B(n) where i XOR 20 = i - 20 starts as 20, 21, 22, 23, 28, 29, 30, 31, 52, ... with successive differences of 1, 1, 1, 5, 1, 1, 1, 21.
for A(2), which is 37, the sequence B(n) where i XOR 36 = i - 36 starts as 36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 100, ... with successive differences of 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 37.

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A057555 (lexicographic ordering).
Rows: A168614(i=1), n>=4.
Cols: A220087(j=2), n>=6.

//program generates values in a table form, row labels of 2^i -1
for i:=1 to 10 do
    m:=2^i - 1;
    m, [ m*2^(n+3) +5 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(3+i-j) + 5;
       if IsPrime(k) then k, "*";
          else k;
       end if;
    end for;
end for;

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 3) + 5 for n>=1.

A242475 a(n) = 2^n + 8.

Original entry on oeis.org

9, 10, 12, 16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832

Offset: 0

Vincenzo Librandi, May 20 2014

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

Cf. A000051, A052548, A062709, A140504, A168614, A153972, A168415, A188165.

Magma
```
[2^n+8: n in [0..40]];
```

Table[2^n + 8, {n, 0, 40}] (* or *) CoefficientList[Series[(9 - 17 x)/((1 - x) (1 - 2 x)),{x, 0, 30}], x]
LinearRecurrence[{3,-2},{9,10},40] (* Harvey P. Dale, May 21 2025 *)

G.f.: (9 - 17*x)/((1 - x)*(1 - 2*x)).
a(n) = 2*a(n-1) - 8 = 3*a(n-1) - 2*a(n-2).
a(n) = A052548(n)+6 = A140504(n)+4 = A153972(n)+2.
E.g.f.: exp(2*x) + 8*exp(x). - Elmo R. Oliveira, Nov 11 2023

A246139 a(n) = 2^n + 10.

Original entry on oeis.org

11, 12, 14, 18, 26, 42, 74, 138, 266, 522, 1034, 2058, 4106, 8202, 16394, 32778, 65546, 131082, 262154, 524298, 1048586, 2097162, 4194314, 8388618, 16777226, 33554442, 67108874, 134217738, 268435466, 536870922, 1073741834, 2147483658, 4294967306

Offset: 0

Vincenzo Librandi, Aug 18 2014

First trisection of A085688. [Bruno Berselli, Aug 19 2014]

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

Cf. Sequences of the form 2^n + k: A000079 (k=0), A000051 (k=1), A052548 (k=2), A062709 (k=3), A140504 (k=4), A168614 (k=5), A153972 (k=6), A168415 (k=7), A242475 (k=8), A188165 (k=9), this sequence (k=10).

Cf. A085688.

Magma
```
[2^n+10: n in [0..40]];
```
Mathematica
```
Table[2^n + 10, {n, 0, 40}]
```

vector(50, n, 2^(n-1)+10) \\ Derek Orr, Aug 18 2014

G.f.: (11 - 21*x)/(1 - 3*x + 2*x^2).
a(n) = A000079(n) + 10.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
E.g.f.: exp(2*x) + 10*exp(x). - Elmo R. Oliveira, Nov 11 2023

A254467 a(n) = 154^n + 702^n + 35*3^n + 5^(n+1) + 6^n + 126.

Original entry on oeis.org

252, 462, 1122, 3432, 12342, 49632, 216342, 1001952, 4863462, 24500352, 127161462, 676195872, 3668030982, 20227217472, 113076824982, 639383508192, 3649985092902, 21003583828992, 121677813214902, 708891056106912, 4149610383537222

Offset: 0

Luciano Ancora, Jan 31 2015

This is the sequence of sixth terms of "fifth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A168614, A254368, A254369, A254370, A254468.

Table[15×4^n+70×2^n+35×3^n+5^(n+1)+6^n+126, {n, 0, 25}] (* Michael De Vlieger, Jan 31 2015 *)
LinearRecurrence[{21,-175,735,-1624,1764,-720},{252,462,1122,3432,12342,49632},30] (* Harvey P. Dale, Jul 16 2018 *)

vector(30, n, n--; 15*4^n + 70*2^n + 35*3^n + 5^(n+1) + 6^n + 126) \\ Colin Barker, Jan 31 2015

G.f.: -6*(21310*x^5 -34383*x^4 +20750*x^3 -5920*x^2 +805*x -42) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)). - Colin Barker, Jan 31 2015

a(n) = 21*a(n-1)-175*a(n-2)+735*a(n-3)-1624*a(n-4)+1764*a(n-5)-720*a(n-6). - Colin Barker, Jan 31 2015

A254468 a(n) = 354^n + 1262^n + 703^n + 155^n + 5*6^n + 7^n + 210.

Original entry on oeis.org

462, 924, 2508, 8646, 35112, 159654, 787968, 4137966, 22807752, 130656534, 772253328, 4683193086, 29012227992, 182964472614, 1171328741088, 7594839621006, 49780643849832, 329318254755894, 2195866174387248, 14741498331453726, 99542297086537272

Offset: 0

Luciano Ancora, Jan 31 2015

This is the sequence of seventh terms of "fifth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A168614, A254368, A254369, A254370, A254467.

Table[35 4^n + 126 2^n + 70 3^n + 15 5^n + 5 6^n + 7^n + 210, {n, 0, 25}] (* Michael De Vlieger, Jan 31 2015 *)
LinearRecurrence[{28,-322,1960,-6769,13132,-13068,5040},{462,924,2508,8646,35112,159654,787968},30] (* Harvey P. Dale, Dec 29 2019 *)

vector(30, n, n--; 35*4^n + 126*2^n + 70*3^n + 15*5^n + 5*6^n + 7^n + 210) \\ Colin Barker, Jan 31 2015

G.f.: -6*(259610*x^6 -461263*x^5 +319473*x^4 -111595*x^3 +20900*x^2 -2002*x +77) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)). - Colin Barker, Jan 31 2015

a(n) = 28*a(n-1) -322*a(n-2) +1960*a(n-3) -6769*a(n-4) +13132*a(n-5) -13068*a(n-6) +5040*a(n-7). - Colin Barker, Jan 31 2015

A248604 Numbers a(n) which are the minimum number of moves needed in a variation of the tower of Hanoi with 4 towers and n disks.

Original entry on oeis.org

1, 3, 5, 9, 13, 21, 37, 69, 133, 261, 517, 1029, 2053, 4101, 8197, 16389, 32773, 65541, 131077, 262149, 524293, 1048581, 2097157, 4194309, 8388613, 16777221, 33554437, 67108869, 134217733, 268435461, 536870917, 1073741829, 2147483653, 4294967301, 8589934597

Offset: 1

Aaron C. Horak, Oct 09 2014

Stefano Spezia, Table of n, a(n) for n = 1..3300
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A168614.

LinearRecurrence[{3,-2},{1,3,5,9,13},40] (* Harvey P. Dale, May 10 2019 *)

a(1)=1; a(2)=3; a(3)=5; a(n) = 5 + 2^(n-2) for n > 3.

G.f.: x*(1 - 2*x^2 - 4*x^4)/((1 - x)*(1 - 2*x)). - Stefano Spezia, May 15 2023

More terms from Harvey P. Dale, May 10 2019

cp's OEIS Frontend

A245318 Numbers k that divide 2^k + 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A254368 a(n) = 5*2^n + 3^n + 15.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254369 a(n) = 15*2^n + 4^n + 5*3^n + 35.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254370 a(n) = 35*2^n + 5*4^n + 15*3^n + 5^n + 70.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

A242475 a(n) = 2^n + 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A246139 a(n) = 2^n + 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A254467 a(n) = 15*4^n + 70*2^n + 35*3^n + 5^(n+1) + 6^n + 126.

Views

A254369 a(n) = 152^n + 4^n + 53^n + 35.

A254370 a(n) = 352^n + 54^n + 15*3^n + 5^n + 70.

A254467 a(n) = 154^n + 702^n + 35*3^n + 5^(n+1) + 6^n + 126.

A254468 a(n) = 354^n + 1262^n + 703^n + 155^n + 5*6^n + 7^n + 210.