A003261 -id:A003261 - cp's OEIS frontend

A366899 Number of prime factors of n*2^n - 1, counted with multiplicity.

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

Offset: 1

Tyler Busby, Oct 26 2023

nonn

The numbers n*2^n-1 are called Woodall (or Riesel) numbers.

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A001222, A003261, A085723, A366898 (divisors), A367006 (without multiplicity).

Table[PrimeOmega[n*2^n - 1], {n, 1, 100}] (* Amiram Eldar, Dec 09 2023 *)

a(n) = bigomega(n*2^n - 1); \\ Michel Marcus, Dec 09 2023

a(n) = bigomega(n*2^n - 1) = A001222(A003261(n)).

A064753 a(n) = n*7^n - 1.

Original entry on oeis.org

6, 97, 1028, 9603, 84034, 705893, 5764800, 46118407, 363182462, 2824752489, 21750594172, 166095446411, 1259557135290, 9495123019885, 71213422649144, 531726889113615, 3954718737782518, 29311444762388081, 216579008522089716, 1595845325952240019, 11729463145748964146

Offset: 1

N. J. A. Sloane, Oct 19 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Index entries for linear recurrences with constant coefficients, signature (15,-63,49).

For a(n)=n*k^n-1 cf. -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), this sequence (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), A064757 (k=11), A064758 (k=12).

Cf. A036293.

[ n*7^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011

k:= 7; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021

Table[n 7^n-1,{n,20}] (* or *) LinearRecurrence[{15,-63,49},{6,97,1028},20] (* Harvey P. Dale, Feb 12 2022 *)

From Alois P. Heinz, Feb 19 2021: (Start)
G.f.: (56*x^2-21*x+1)/((x-1)*(7*x-1)^2).
a(n) = A036293(n) - 1. (End)
From Elmo R. Oliveira, May 05 2025: (Start)
E.g.f.: 1 + exp(x)*(7*x*exp(6*x) - 1).
a(n) = 15*a(n-1) - 63*a(n-2) + 49*a(n-3) for n > 3. (End)

A064756 a(n) = n*10^n - 1.

Original entry on oeis.org

9, 199, 2999, 39999, 499999, 5999999, 69999999, 799999999, 8999999999, 99999999999, 1099999999999, 11999999999999, 129999999999999, 1399999999999999, 14999999999999999, 159999999999999999, 1699999999999999999, 17999999999999999999, 189999999999999999999, 1999999999999999999999

Offset: 1

N. J. A. Sloane, Oct 19 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. for a(n) = n*k^n - 1: -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), this sequence (k=10), A064757 (k=11), A064758 (k=12).

Cf. A064748, A126431.

[ n*10^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011

k:= 10; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021

Array[# 10^# - 1 &, 18] (* Michael De Vlieger, Jan 14 2020 *)

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(100*x^2 - 10*x - 9)/((x - 1)*(10*x - 1)^2).
E.g.f.: 1 + exp(x)*(10*x*exp(9*x) - 1).
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3) for n > 3.
a(n) = A126431(n) - 1 = A064748(n) - 2. (End)

A064757 a(n) = n*11^n - 1.

Original entry on oeis.org

10, 241, 3992, 58563, 805254, 10629365, 136410196, 1714871047, 21221529218, 259374246009, 3138428376720, 37661140520651, 448795257871102, 5316497670165373, 62658722541234764, 735195677817154575, 8592599484487994106, 100078511642860166657, 1162022718519876379528

Offset: 1

N. J. A. Sloane, Oct 19 2001

Conjecture: satisfies a linear recurrence having signature (23,-143,121). - Harvey P. Dale, May 12 2019

This conjecture is true since a(n) - a(n-1) yields the recurrence 1 + 10*n + 11*n*a(n-1) - (n-1)*a(n) = 0 with polynomial coefficients in n. - Georg Fischer, Feb 19 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Index entries for linear recurrences with constant coefficients, signature (23,-143,121).

Cf. for a(n) = n*k^n - 1: -A000012(k=0), A001477(k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), this sequence (k=11), A064758 (k=12).

Cf. A064749.

[n*11^n - 1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011

k:= 11; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021

Table[n*11^n-1,{n,20}] (* Harvey P. Dale, May 12 2019 *)

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(121*x^2 - 11*x - 10)/((x - 1)*(11*x - 1)^2).
E.g.f.: 1 + exp(x)*(11*x*exp(10*x) - 1).
a(n) = 23*a(n-1) - 143*a(n-2) + 121*a(n-3) for n > 3.
a(n) = A064749(n) - 2. (End)

A064758 a(n) = n*12^n - 1.

Original entry on oeis.org

11, 287, 5183, 82943, 1244159, 17915903, 250822655, 3439853567, 46438023167, 619173642239, 8173092077567, 106993205379071, 1390911669927935, 17974858503684095, 231105323618795519, 2958148142320582655, 37716388814587428863, 479219999055934390271, 6070119988041835610111, 76675199848949502443519

Offset: 1

N. J. A. Sloane, Oct 19 2001

Harry J. Smith, Table of n, a(n) for n = 1..150
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Index entries for linear recurrences with constant coefficients, signature (25,-168,144).

Cf. for a(n) = n*k^n - 1: -A000012(k=0), A001477(k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), A064757 (k=11), this sequence (k=12).

Cf. A064750.

[n*12^n - 1: n in [1..30]]; // Vincenzo Librandi, Jun 21 2018

CoefficientList[Series[(11 + 12 x - 144 x^2) / ((1 - 12 x)^2 (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 21 2018 *)

a(n) = { n*12^n - 1 } \\ Harry J. Smith, Sep 24 2009

G.f.: x*(11 + 12*x - 144*x^2)/((1 - 12*x)^2*(1 - x)). - Vincenzo Librandi, Jun 21 2018
From Elmo R. Oliveira, Sep 07 2024: (Start)
E.g.f.: 1 + exp(x)*(12*x*exp(11*x) - 1).
a(n) = 25*a(n-1) - 168*a(n-2) + 144*a(n-3) for n > 3.
a(n) = A064750(n) - 2. (End)

A366898 Number of divisors of n*2^n - 1, the Woodall (or Riesel) numbers.

Original entry on oeis.org

1, 2, 2, 6, 4, 2, 4, 4, 4, 4, 6, 4, 12, 10, 8, 48, 8, 4, 8, 4, 16, 16, 8, 8, 4, 8, 8, 16, 24, 2, 8, 4, 8, 32, 8, 32, 4, 8, 4, 24, 16, 8, 32, 8, 16, 24, 40, 16, 16, 8, 8, 16, 24, 8, 16, 8, 16, 6, 32, 8, 8, 16, 8, 512, 48, 16, 12, 48, 16, 8, 8, 4, 24, 16, 2, 256

Offset: 1

Tyler Busby, Oct 26 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A000005, A003261, A002234, A242273, A063515, A056821, A367006, A366899.

Table[DivisorSigma[0, n*2^n - 1], {n, 1, 100}] (* Amiram Eldar, Dec 11 2023 *)

a(n) = numdiv(n*2^n - 1); \\ Amiram Eldar, Dec 11 2023

a(n) = sigma0(n*2^n - 1) = A000005(A003261(n)).

A367003 a(n) is the largest prime factor of n*2^n-1.

Original entry on oeis.org

1, 7, 23, 7, 53, 383, 179, 89, 271, 3413, 2503, 2137, 59, 367, 1433, 41, 15803, 59729, 26423, 11161, 1559, 12611, 9187523, 127867, 119837257, 11527, 2360833, 43969, 2339, 32212254719, 257503, 616318177, 260587, 127873, 682902239, 44939, 69660839431, 1185617

Offset: 1

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A003261, A006530, A005420, A367002, A367004, A366899.

a[n_] := FactorInteger[n*2^n - 1][[-1, 1]]; Array[a, 40] (* Amiram Eldar, Oct 29 2024 *)

a(n) = A006530(A003261(n)).

A367006 Number of distinct prime factors of n*2^n - 1.

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Oct 31 2023

nonn

The numbers n*2^n-1 are called Woodall (or Riesel) numbers.

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A003261, A001221, A366898, A366899.

Table[PrimeNu[n*2^n - 1], {n, 1, 100}] (* Amiram Eldar, Dec 11 2023 *)

a(n) = omega(n*2^n - 1); \\ Amiram Eldar, Dec 11 2023

a(n) = omega(n*2^n - 1) = A001221(A003261(n)).

A236752 Primes of the form k*2^(k-1) - 1.

Original entry on oeis.org

3, 11, 31, 79, 191, 5119, 245759, 524287, 1114111, 3758096383, 1618481116086271, 653980173926178609468673073657929531391, 5359447279004780799548150067050349330431

Offset: 1

Gerasimov Sergey, Jan 30 2014

nonn

Primes in A087323.
Corresponding values of k: 2, 3, 4, 5, 6, 10, 15, 16, 17, 28, 46, 123, ...
The values of k-1 are listed in A230769. - Jeppe Stig Nielsen, Oct 16 2019

			79 is in this sequence because it is prime and for k = 5, k*2^(k-1) - 1 = 5*2^(5-1) - 1 = 79.

Cf. A002054, A003261, A196273, A230769.

More terms and corrections of terms and comments by Ralf Stephan, Feb 03 2014

A367002 a(n) is the smallest prime factor of n*2^n-1.

Original entry on oeis.org

7, 23, 3, 3, 383, 5, 23, 17, 3, 3, 23, 5, 5, 7, 3, 3, 79, 13, 1879, 13, 3, 3, 47, 7, 229, 5, 3, 3, 32212254719, 263, 223, 5, 3, 3, 5, 73, 17, 1217, 3, 3, 6709, 29, 7, 71, 3, 3, 11, 97, 47, 228713, 3, 3, 5, 37, 5, 7, 3, 3, 9377, 11, 13, 479, 3, 3, 41, 5, 13, 137

Offset: 2

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 2..884 (terms 2..325 from Robert Israel)

Cf. A003261, A020639, A049479, A367003, A367004, A366899.

f:= n -> min(numtheory:-factorset(n*2^n-1)):
map(f, [$2..100]); # Robert Israel, Nov 08 2023

Table[FactorInteger[n*2^n-1][[1,1]], {n,2,69}] (* Paul F. Marrero Romero, Dec 17 2023 *)

a(n) = A020639(A003261(n)).

a(n) = 3 iff n == 4 or 5 (mod 6). - Robert Israel, Nov 08 2023

cp's OEIS Frontend

A366899 Number of prime factors of n*2^n - 1, counted with multiplicity.

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A064753 a(n) = n*7^n - 1.

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Magma

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A064756 a(n) = n*10^n - 1.

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Magma

Maple

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A064757 a(n) = n*11^n - 1.

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Magma

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A064758 a(n) = n*12^n - 1.

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A366898 Number of divisors of n*2^n - 1, the Woodall (or Riesel) numbers.

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A367003 a(n) is the largest prime factor of n*2^n-1.

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A367006 Number of distinct prime factors of n*2^n - 1.

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