A024023 -id:A024023 - cp's OEIS frontend

A295500 a(n) = phi(3^n-1), where phi is Euler's totient function (A000010).

1, 4, 12, 32, 110, 288, 1092, 2560, 9072, 26400, 84700, 165888, 797160, 2384928, 6019200, 15728640, 64533700, 141087744, 580765248, 1246080000, 4823425152, 14758128000, 46070066188, 85996339200, 385087175000, 1270928131200, 3474144608256, 8810420097024

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690
Eric Weisstein's World of Mathematics, Totient Function
Wikipedia, Euler's totient function

Cf. A000010, A024023, A027385.

phi(k^n-1): A053287 (k=2), this sequence (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

EulerPhi[3^Range[30] - 1] (* Paolo Xausa, Jun 18 2024 *)

PARI
```
{a(n) = eulerphi(3^n-1)}
```

a(n) = n*A027385(n).

a(n) = A000010(A024023(n)). - Michel Marcus, Jun 18 2024

A074477 Largest prime factor of 3^n - 1.

Original entry on oeis.org

2, 2, 13, 5, 11, 13, 1093, 41, 757, 61, 3851, 73, 797161, 1093, 4561, 193, 34511, 757, 363889, 1181, 368089, 3851, 1001523179, 6481, 391151, 797161, 8209, 16493, 20381027, 4561, 4404047, 21523361, 2413941289, 34511, 2664097031, 530713

Offset: 1

Rick L. Shepherd, Aug 23 2002

nonn

			3^7 - 1 = 2186 = 2*1093, so a(7) = 1093.

Table of n, a(n) for n = 1..690
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006530 (largest prime factor), A024023 (3^n-1).

Cf. A074476 (largest prime factor of 3^n + 1), A005420 (largest prime factor of 2^n - 1), A074479 (largest prime factor of 5^n - 1).

[Maximum(PrimeDivisors(3^n-1)): n in [1..40]]; // Vincenzo Librandi, Aug 23 2013

A074477 := proc(n)
        A006530( 3^n-1) ;
end proc: # R. J. Mathar, Jul 18 2015
# alternative:
a:= n-> max(seq(i[1], i=ifactors(3^n-1)[2])):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 18 2015

Table[FactorInteger[3^n - 1] [[-1, 1]], {n, 40}] (* Vincenzo Librandi, Aug 23 2013 *)

for(n=1,40, v=factor(3^n-1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A024023(n)). - Michel Marcus, Jul 18 2015

Terms to a(100) in b-file from Vincenzo Librandi, Aug 23 2013
a(101)-a(660) in b-file from Amiram Eldar, Feb 01 2020
a(661)-a(690) in b-file from Max Alekseyev, May 22 2022

A115099 a(0)=4, a(n) = 3*a(n-1) - 4.

Original entry on oeis.org

4, 8, 20, 56, 164, 488, 1460, 4376, 13124, 39368, 118100, 354296, 1062884, 3188648, 9565940, 28697816, 86093444, 258280328, 774840980, 2324522936, 6973568804, 20920706408, 62762119220, 188286357656, 564859072964, 1694577218888, 5083731656660, 15251194969976

Offset: 0

Miklos Kristof, Mar 02 2006

A tetrahedron has 4 faces. Cut every corner so that we get triangular faces; the resulting polyhedron has 8 faces. Repeating this procedure gives polyhedra with 4, 8, 20, 56, etc. faces.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931. - Vladimir Joseph Stephan Orlovsky, Dec 25 2008

[2*3^n+2: n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

Maple
```
seq(2*3^i+2,i=0..30);
```

a=4;lst={a};Do[a=a*3-4;AppendTo[lst,a],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)

Vec(4*(1-2*x)/((1-x)*(1-3*x)) + O(x^30)) \\ Colin Barker, May 31 2016

a(n) = 2*3^n + 2.
From Colin Barker, May 31 2016: (Start)
a(n) = 4*a(n-1)-3*a(n-2) for n>1.
G.f.: 4*(1-2*x) / ((1-x)*(1-3*x)).
(End)
E.g.f.: 2*(1 + exp(2*x))*exp(x). - Ilya Gutkovskiy, May 31 2016
a(n) = 4 * A007051(n). - Alois P. Heinz, Jun 26 2023

A257833 Table T(k, n) of smallest bases b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^k), read by antidiagonals.

Original entry on oeis.org

5, 8, 9, 7, 26, 17, 18, 57, 80, 33, 3, 18, 182, 242, 65, 19, 124, 1047, 1068, 728, 129, 38, 239, 1963, 1353, 1068, 2186, 257, 28, 158, 239, 27216, 34967, 32318, 6560, 513, 28, 333, 4260, 109193, 284995, 82681, 110443, 19682, 1025, 14, 42, 2819, 15541, 861642, 758546, 2387947, 280182, 59048, 2049

Offset: 2

Felix Fröhlich, May 10 2015

			T(3, 5) = 124, since prime(5) = 11 and the smallest b such that b^10 == 1 (mod 11^3) is 124.
Table starts
  k\n|    1     2       3        4       5       6         7
  ---+----------------------------------------------------------
   2 |    5     8       7       18       3      19        38 ...
   3 |    9    26      57       18     124     239       158 ...
   4 |   17    80     182     1047    1963     239      4260 ...
   5 |   33   242    1068     1353   27216  109193     15541 ...
   6 |   65   728    1068    34967  284995  861642    390112 ...
   7 |  129  2186   32318    82681  758546 6826318  21444846 ...
   8 |  257  6560  110443  2387947 9236508 6826318 112184244 ...
   9 |  513 19682  280182 14906455 ....
  10 | 1025 59048 3626068 ....
  ...

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Column 1 of table is A000051.
Column 2 of table is A024023 (with offset 2).
Column 3 of table is A034939 (with offset 2).
Rows k=2-10 give: A039678, A249275, A353937, A353938, A353939, A353940, A353941, A353942, A353943.

for(k=2, 10, forprime(p=2, 25, b=2; while(Mod(b, p^k)^(p-1)!=1, b++); print1(b, ", ")); print(""))

T(k,n) = my(p=prime(n), v=List([2])); if(n==1, return(2^k+1)); for(i=1, k, w=List([]); for(j=1, #v, forstep(b=v[j], p^i-1, p^(i-1), if(Mod(b, p^i)^p==b, listput(w, b)))); v=Vec(w)); vecmin(v); \\ Jinyuan Wang, May 17 2022

from itertools import count, islice
from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A257833_T(n,k): return 2**k+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**k,True)[1])
def A257833_gen(): # generator of terms
    yield from (A257833_T(n,i-n+2) for i in count(1) for n in range(i,0,-1))
A257833_list = list(islice(A257833_gen(),50)) # Chai Wah Wu, May 17 2022

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 20, 26, 28, 30, 34, 38, 42, 46, 52, 56, 62, 68, 72, 80, 82, 84, 88, 92, 96, 100, 106, 110, 116, 122, 126, 134, 140, 144, 152, 160, 164, 170, 176, 180, 188, 194, 198, 204, 212, 216, 224, 232, 242, 244, 246, 250, 254, 258, 262, 268, 272, 278, 284, 288, 296, 302, 306, 314, 322, 326, 332, 338, 342, 350, 356, 360

Offset: 0

Antti Karttunen, Sep 11 2016

Antti Karttunen, Table of n, a(n) for n = 0..6250

Cf. A004128, A024023, A053735, A054861, A261231 (left inverse), A261233, A276622, A276624, A276603 (terms divided by 2), A276604 (first differences).
Cf. A179016, A219648, A219666, A255056, A259934, A276573, A276583, A276613 for similar constructions.
Cf. also A263273.

(define (A276623 n) (A276624 (A276622 n)))

a(n) = A276624(A276622(n)).
Other identities. For all n >= 0:
A261231(a(n)) = n.
a(A261233(n)) = A024023(n) = 3^n - 1.

A366578 Sum of the divisors of 3^n+1.

Original entry on oeis.org

3, 7, 18, 56, 126, 434, 1332, 3836, 10476, 42560, 109926, 315112, 816732, 2790074, 8906760, 30220288, 64570086, 229156928, 706911048, 2034690952, 5357742012, 21838961760, 56496274632, 164750562956, 456919958880, 1517043139136, 4661686010664, 16489453890560

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

			a(4)=126 because 3^4+1 has divisors {1, 2, 41, 82}.

Max Alekseyev, Table of n, a(n) for n = 0..691

Cf. A000203, A002592, A024023, A057935, A057941, A074476, A075708, A274909, A366576, A366577, A366579, A366580

a:=n->numtheory[sigma](3^n+1):
seq(a(n), n=0..100);

DivisorSigma[1,3^Range[0,30]+1] (* Paolo Xausa, Oct 15 2023 *)

a(n) = sigma(3^n+1) = A000203(A034472(n)).

A024140 a(n) = 12^n - 1.

Original entry on oeis.org

0, 11, 143, 1727, 20735, 248831, 2985983, 35831807, 429981695, 5159780351, 61917364223, 743008370687, 8916100448255, 106993205379071, 1283918464548863, 15407021574586367, 184884258895036415

Offset: 0

N. J. A. Sloane

In base 12 these are 0, B, BB, BBB, ... . - David Rabahy, Dec 12 2016

Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A001021, A016125, A178248.

Cf. Similar sequences of the type k^n-1: A000004 (k=1), A000225 (k=2), A024023 (k=3), A024036 (k=4), A024049 (k=5), A024062 (k=6), A024075 (k=7), A024088 (k=8), A024101 (k=9), A002283 (k=10), A024127 (k=11), this sequence (k=12).

12^Range[0,20]-1 (* or *) LinearRecurrence[{13,-12},{0,11},20] (* Harvey P. Dale, Feb 01 2019 *)

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-12*x) - 1/(1-x).
E.g.f.: exp(12*x) - exp(x). (End)
a(n) = 12*a(n-1) + 11 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{i=1..n} 11^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
From Elmo R. Oliveira, Dec 15 2023: (Start)
a(n) = 13*a(n-1) - 12*a(n-2) for n>1.
a(n) = A001021(n)-1 = A178248(n)-2.
a(n) = 11*(A016125(n) - 1)/12. (End)

A048328 Numbers that are repdigits in base 3.

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 26, 40, 80, 121, 242, 364, 728, 1093, 2186, 3280, 6560, 9841, 19682, 29524, 59048, 88573, 177146, 265720, 531440, 797161, 1594322, 2391484, 4782968, 7174453, 14348906, 21523360, 43046720, 64570081, 129140162, 193710244, 387420488, 581130733

Offset: 0

Patrick De Geest, Feb 15 1999

Case for base 2 see A000225: 2^n - 1.

If the sequence b(n) represents the number of paths of length n, n >= 1, starting at node 1 and ending at nodes 1, 2, 3 and 4 on the path graph P_5 then a(n-1) = b(n) - 1. - Johannes W. Meijer, May 29 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for 3-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Bisections: A024023 and A003462.

Cf. A000225, A010785, A028987, A028988, A033016, A038754, A068911, A124302, A214369.

nmax := 35; a(0) := 0: for n from 1 to nmax do a(2*n) := a(2*n-2) + 2*3^(n-1); od: a(1) := 1: for n from 1 to nmax do a(2*n+1) := 1*a(2*n-1) + 3^n; od: seq(a(n), n=0..nmax);
# End program 1
with(GraphTheory): G := PathGraph(5): A:= AdjacencyMatrix(G): nmax := nmax; for n from 1 to nmax+1 do B(n) := A^n; b(n) := add(B(n)[1, k], k=1..4); a1(n-1) := b(n)-1; od: seq(a1(n), n=0..nmax);
# End program 2
# From Johannes W. Meijer, May 29 2010, revised Sep 23 2012
# third Maple program:
a:= n->(<<0|1>, <-3|4>>^iquo(n, 2, 'r').`if`(r=0, <<0, 2>>, <<1, 4>>))[1, 1]:
seq (a(n), n=0..60);  # Alois P. Heinz, Sep 23 2012

Rest[FromDigits[#, 3]&/@Flatten[Table[{PadRight[{1}, n, 1], PadRight[{2}, n, 2]}, {n, 0, 20}], 1]] (* Harvey P. Dale, Feb 03 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -3,0,4,0]^n*[0;1;2;4])[1,1] \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (2*x^2+x)/(1-4*x^2+3*x^4). - Alois P. Heinz, Sep 23 2012
Sum_{n>=1} 1/a(n) = 3 * A214369 = 2.04646050781571420028... - Amiram Eldar, Jan 21 2022
a(n) = (3^(n/2)*(sqrt(3) + 2 - (-1)^n*(sqrt(3) - 2)) - 3 - (-1)^n)/4. - Stefano Spezia, Feb 18 2022

A227048 Irregular triangle read by rows: row n, for n >= 0, lists the nonnegative differences 3^n - 2^m, m >= 0, in increasing order.

Original entry on oeis.org

0, 1, 2, 1, 5, 7, 8, 11, 19, 23, 25, 26, 17, 49, 65, 73, 77, 79, 80, 115, 179, 211, 227, 235, 239, 241, 242, 217, 473, 601, 665, 697, 713, 721, 725, 727, 728, 139, 1163, 1675, 1931, 2059, 2123, 2155, 2171, 2179, 2183, 2185, 2186, 2465, 4513, 5537, 6049, 6305

Offset: 0

Reinhard Zumkeller, Jun 29 2013

A020914(n) = length of n-th row;
T(n,1) = A056577(n);
T(n,A098294(n)) = A001047(n);
T(n,A020914(n)) = A024023(n);
T(n,k) = A196486(n,A020914(n)-k) for n > 0, k = 1..A056576(n).

			Initial rows:
0:  0
1:  1,2
2:  1,5,7,8
3:  11,19,23,25,26 (= 27-16, 27-8, 27-4, 27-2, 27-1)
4:  17,49,65,73,77,79,80
5:  115,179,211,227,235,239,241,242
6:  217,473,601,665,697,713,721,725,727,728
7:  139,1163,1675,1931,2059,2123,2155,2171,2179,2183,2185,2186
8:  2465,4513,5537,6049,6305,6433,6497,6529,6545,6553,6557,6559,6560
...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Cf. A000079, A000244, A192111.

See also A001047, A020914, A024023, A056576, A056577, A098294, A196486.

a227048 n k = a227048_tabf !! n !! (k-1)
a227048_row n = a227048_tabf !! n
a227048_tabf = map f a000244_list  where
   f x = reverse $ map (x -) $ takeWhile (<= x) a000079_list

Definition revised by N. J. A. Sloane, Oct 11 2019

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

Original entry on oeis.org

3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151

Offset: 0

Philippe Deléham, Feb 16 2014

a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020
First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Radius.
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* Start from Eric W. Weisstein, Mar 13 2018 *)
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = A000244(n+1) + A003462(n).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
a(n) = A199109(n) - 1.
a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018
From Klaus Purath, Apr 13 2020: (Start)
a(n) = A057198(n+1) + A024023(n).
a(n) = A029858(n+2) - A024023(n).
a(n) = A052919(n+1) + A029858(n+1).
a(n) = (A000244(n+1) + A171498(n))/2.
a(n) = 7*A003462(n) + 3.
a(n) = A116952(n) + 2. (End)
a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020
E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

cp's OEIS Frontend

A295500 a(n) = phi(3^n-1), where phi is Euler's totient function (A000010).

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A074477 Largest prime factor of 3^n - 1.

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A115099 a(0)=4, a(n) = 3*a(n-1) - 4.

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A257833 Table T(k, n) of smallest bases b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^k), read by antidiagonals.

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A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

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A366578 Sum of the divisors of 3^n+1.

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A024140 a(n) = 12^n - 1.

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A048328 Numbers that are repdigits in base 3.