A067886 -id:A067886 - cp's OEIS frontend

A283364 Numbers m such that both numbers 2^m +- 1 have at most 2 distinct prime factors.

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 31, 61, 101, 127, 167, 199, 347

Offset: 1

Vladimir Shevelev, Mar 06 2017

If a(n) > 9 then a(n) is prime. Proof: If k = 2*m > 9 then 2^(2*m)-1 has at least 3 factors; being 3, (2^m - 1) / 3 and 2^m + 1 which excludes even numbers > 9.
If k = 2*m + 1 > 9 is not prime then k = p*q, q, p > 3 so 2^(p*q) + 1 is divisible by 3, 2^p + 1 and 2^q + 1. If p = q then 2^(p^2) + 1 is divisible by 3, 2^p + 1 and (2^p^2 + 1) / (2^p + 1) > 2^p + 1. Which excludes odd composite numbers > 9 and completes the proof. [comments reworded by David A. Corneth, Nov 23 2019]
Any further terms are > 1122. - Lucas A. Brown, Oct 21 2024

Cf. A001221, A046799, A046800, A067886.

Select[Range@ 200, Times @@ Boole@ Map[PrimeNu@ # <= 2 &, 2^# + {-1, 1}] == 1 &] (* Michael De Vlieger, Mar 06 2017 *)
Select[Range[350],Max[PrimeNu[2^#+{1,-1}]]<3&] (* Harvey P. Dale, Dec 23 2017 *)

isok(n) = omega(2^n+1)<=2 && omega(2^n-1)<=2;
for(n=1, 347, if(isok(n)==1, print1(n,", "))); \\ Indranil Ghosh, Mar 06 2017

More terms from Peter J. C. Moses, Mar 06 2017

A337811 Numbers k such that the number of distinct prime factors of 2^k - 1 is less than the corresponding count for 2^k + 1.

Original entry on oeis.org

1, 5, 7, 13, 17, 19, 25, 26, 31, 34, 35, 37, 38, 41, 46, 49, 59, 61, 62, 65, 67, 77, 78, 83, 85, 89, 91, 93, 97, 98, 103, 107, 109, 118, 122, 123, 125, 127, 131, 133, 134, 137, 139, 143, 145, 147, 149, 153, 157, 170, 173, 175, 177, 185, 186, 189, 193, 194, 195

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A046799, A046800, A067886, A337810, A337812, A337813.

Select[Range[200],PrimeNu[2^#-1]Harvey P. Dale, Nov 04 2023 *)

PARI
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for(n=1,200,if(omega(2^n-1)
				
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A337813 Numbers k such that the number of distinct prime factors of 2^k - 1 is greater than the corresponding count for 2^k + 1.

Original entry on oeis.org

4, 8, 10, 12, 16, 20, 22, 24, 28, 30, 32, 36, 39, 40, 43, 44, 45, 48, 50, 52, 55, 56, 58, 60, 63, 64, 66, 68, 70, 72, 75, 76, 79, 80, 84, 87, 88, 90, 92, 94, 96, 99, 100, 102, 104, 106, 108, 110, 112, 116, 117, 119, 120, 124, 126, 128, 132, 135, 136, 140, 144

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A046799, A046800, A067886, A337810, A337811, A337812.

for(n=1,150,if(omega(2^n-1)>omega(2^n+1),print1(n,", ")))

A309942 Numbers k such that 2^k - 1 and 2^k + 1 have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

2, 10, 11, 14, 21, 23, 29, 39, 47, 50, 53, 55, 63, 71, 73, 74, 75, 82, 86, 95, 101, 105, 113, 115, 121, 142, 147, 150, 167, 169, 179, 181, 182, 190, 199, 203, 209, 233, 235, 253, 277, 285, 303, 307, 311, 317, 335, 337, 339, 342, 343, 347, 349, 353, 355, 358

Offset: 1

Hugo Pfoertner, Aug 24 2019

nonn

			a(1) = 2: 2^2 - 1 = 3 and 2^2 + 1 are both prime,
a(2) = 10: 2^10 - 1 = 1023 = 3 * 11 * 31 and 2^10 + 1 = 1025 = 5^2 * 41 both have 3 prime factors.

Cf. A000051, A000225, A046051, A054992, A067886.

[m:m in [2..400]| &+[p[2]: p in Factorization(2^m-1)] eq &+[p[2]: p in Factorization(2^m+1)]]; // Marius A. Burtea, Aug 24 2019

Select[Range[200], PrimeOmega[2^# - 1 ] == PrimeOmega[2^# + 1 ] &] (* Amiram Eldar, Aug 24 2019 *)

for(k=1, 209, my(f=bigomega(2^k-1),g=bigomega(2^k+1));if(f==g,print1(k,", ")))

More terms from Amiram Eldar, Aug 24 2019

cp's OEIS Frontend

A283364 Numbers m such that both numbers 2^m +- 1 have at most 2 distinct prime factors.

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A337811 Numbers k such that the number of distinct prime factors of 2^k - 1 is less than the corresponding count for 2^k + 1.

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A337813 Numbers k such that the number of distinct prime factors of 2^k - 1 is greater than the corresponding count for 2^k + 1.

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A309942 Numbers k such that 2^k - 1 and 2^k + 1 have the same number of prime factors, counted with multiplicity.

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