A366899 -id:A366899 - cp's OEIS frontend

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

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)).

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

A367008 Number of prime factors of n*2^n + 1, counted with multiplicity.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 31 2023

nonn

The numbers n*2^n+1 are called Cullen numbers.

Amiram Eldar, Table of n, a(n) for n = 0..865

Cf. A002064, A001222, A366899, A367007.

Table[PrimeOmega[n*2^n + 1], {n, 0, 100}] (* Amiram Eldar, Jan 06 2024 *)

a(n) = bigomega(n*2^n + 1); \\ Amiram Eldar, Jan 06 2024

a(n) = bigomega(n*2^n + 1) = A001222(A002064(n)).

cp's OEIS Frontend

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

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A367008 Number of prime factors of n*2^n + 1, counted with multiplicity.

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