A024101 -id:A024101 - cp's OEIS frontend

A366660 Number of distinct prime divisors of 9^n - 1.

1, 2, 3, 3, 3, 5, 3, 5, 6, 5, 5, 7, 3, 6, 8, 6, 6, 9, 5, 7, 8, 8, 4, 12, 7, 6, 11, 9, 7, 12, 6, 7, 10, 9, 8, 12, 6, 8, 12, 11, 6, 14, 4, 12, 16, 7, 8, 15, 10, 12, 13, 9, 6, 15, 11, 14, 13, 10, 5, 18, 5, 10, 16, 8, 9, 15, 6, 13, 13, 15, 7, 19, 7, 10, 19, 13, 11

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024101, A001221, A057952, A366661, A366662, A366663.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366651, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(9^n - 1), ", "))

a(n) = omega(9^n-1) = A001221(A024101(n)).

a(n) = A133801(2*n) = A133801(n) + A366580(n) - 1. - Max Alekseyev, Jan 07 2024

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)

A173952 a(1)=32 and, for n > 1, a(n) = 9*a(n-1) + 32.

Original entry on oeis.org

32, 320, 2912, 26240, 236192, 2125760, 19131872, 172186880, 1549681952, 13947137600, 125524238432, 1129718145920, 10167463313312, 91507169819840, 823564528378592, 7412080755407360, 66708726798666272, 600378541187996480

Offset: 1

John W. Layman, Mar 03 2010

It appears that all terms of this sequence are also terms of A173951.

Also, it appears that a(n) has the base-3 representation 1,0,2^(2n-2),1,2 where 2^k denotes k consecutive 2's.

Vincenzo Librandi, Table of n, a(n) for n = 1..400
Index entries for linear recurrences with constant coefficients, signature (10, -9).

Cf. A173951.

[4*(9^n-1): n in [1..20]]; // Vincenzo Librandi, Jul 05 2011

NestList[9#+32&,32,20] (* Harvey P. Dale, May 27 2012 *)

From R. J. Mathar, Mar 04 2010: (Start)
a(n) = 4*(9^n-1) = 4*A024101(n) = 10*a(n-1) - 9*a(n-2).
G.f.: 32*x/ ((9*x-1) * (x-1)). (End)

A363228 Exponent of 4 in 9^n - 1.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 1

Ruud H.G. van Tol, May 21 2023

Not the same as A147648-without-zeros.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007814, A090739, A147648, A244415.

Cf. A024101, A235127.

a[n_] := IntegerExponent[2*n, 4] + 1; Array[a, 100] (* Amiram Eldar, May 22 2023 *)

PARI
```
a(n) = valuation(2*n, 4) + 1;
```

def A363228(n): return (~n&n-1).bit_length()+3>>1 # Chai Wah Wu, Jul 09 2023

a(n) = floor(A090739(n)/2).
a(n) = A244415(n) + 1.
a(n) = A235127(A024101(n)). - Michel Marcus, May 21 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/3. - Amiram Eldar, Jul 13 2023
Conjecture: a(n) = A235127(A000045(6*n)), all other 4-adic 6-sections A235127(A000045(.))=0. - R. J. Mathar, Jun 28 2025

A122760 Triangle read by rows: t(n,m) = 23^m(n mod 2).

Original entry on oeis.org

0, 2, 6, 0, 0, 0, 2, 6, 18, 54, 0, 0, 0, 0, 0, 2, 6, 18, 54, 162, 486, 0, 0, 0, 0, 0, 0, 0, 2, 6, 18, 54, 162, 486, 1458, 4374, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Roger L. Bagula, Sep 21 2006

A Cantor-based power of 3 triangular array.

			0
2, 6
0, 0, 0
2, 6, 18, 54
0, 0, 0, 0, 0
2, 6, 18, 54, 162, 486
0, 0, 0, 0, 0, 0, 0
2, 6, 18, 54, 162, 486, 1458, 4374

Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Dover, New York, 1978, 57-58

Row sums: see A024101.

b[n_] := 2*Mod[n, 2] T2[n_, m_] := 3^n*b[m] b0 = Table[Table[T2[n, m], {n, 0, m}], {m, 0, 10}]; Flatten[b0] MatrixForm[b0]

Edited and corrected by N. J. A. Sloane, May 29 2009

cp's OEIS Frontend

A366660 Number of distinct prime divisors of 9^n - 1.

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

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A173952 a(1)=32 and, for n > 1, a(n) = 9*a(n-1) + 32.

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A363228 Exponent of 4 in 9^n - 1.

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A122760 Triangle read by rows: t(n,m) = 2*3^m*(n mod 2).

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A122760 Triangle read by rows: t(n,m) = 23^m(n mod 2).