A054992 -id:A054992 - cp's OEIS frontend

A366687 Number of prime factors of 11^n + 1 (counted with multiplicity).

1, 3, 2, 5, 2, 4, 4, 4, 3, 7, 3, 7, 4, 6, 5, 8, 3, 6, 5, 7, 4, 7, 4, 7, 7, 6, 3, 10, 6, 6, 6, 7, 4, 13, 6, 11, 7, 5, 4, 11, 5, 6, 9, 5, 6, 13, 6, 7, 5, 8, 6, 11, 3, 7, 9, 13, 7, 12, 6, 7, 8, 6, 4, 13, 3, 10, 8, 9, 7, 14, 8, 6, 10, 8, 8, 13, 6, 12, 12, 7, 10

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A001222, A057934, A062308, A366686, A366688, A366689, A366690.

Cf. A054992, A057941, A057940, A057939, A057938, A057937, A057936, A057935, A057934, A366713.

Mathematica
```
PrimeOmega[11^Range[70]+1]
```
PARI
```
a(n)=bigomega(11^n+1)
```

a(n) = bigomega(11^n+1) = A001222(A034524(n)).

A001269 Table T(n,k) in which n-th row lists prime factors of 2^n + 1 (n >= 0), with repetition.

Original entry on oeis.org

2, 3, 5, 3, 3, 17, 3, 11, 5, 13, 3, 43, 257, 3, 3, 3, 19, 5, 5, 41, 3, 683, 17, 241, 3, 2731, 5, 29, 113, 3, 3, 11, 331, 65537, 3, 43691, 5, 13, 37, 109, 3, 174763, 17, 61681, 3, 3, 43, 5419, 5, 397, 2113, 3, 2796203, 97, 257, 673, 3, 11, 251, 4051

Offset: 0

N. J. A. Sloane

Rows have irregular lengths.

The length of row n is A054992(n).

			Triangle begins:
  2;
  3;
  5;
  3,3,17;
  3,11;
  5,13;
  3,43;
  257;
  ...

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Max Alekseyev, Rows n = 0..1122, flattened (rows 0..500 from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
S. S. Wagstaff, Jr., The Cunningham Project
Chai Wah Wu, Tables from the Cunningham Project in machine-readable JSON format.

Cf. A060444 (factors w/o repetition), A054992 (row lengths).

repeat[{p_, e_}] := Table[p, {e}]; row[n_] := repeat /@ FactorInteger[2^n + 1] // Flatten; Table[row[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Jul 13 2012 *)

apply( A001269_row(n)=concat(apply(f->vector(f[2],i,f[1]), Col(factor(2^n+1))~)), [0..19]) \\ M. F. Hasler, Nov 19 2018

A337810 Numbers k such that the number of prime factors, counted with multiplicity, of 2^k - 1 is less than the corresponding count for 2^k + 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 25, 26, 27, 31, 33, 34, 35, 37, 38, 41, 45, 46, 49, 51, 57, 59, 61, 62, 65, 67, 69, 77, 78, 81, 83, 85, 89, 91, 93, 97, 98, 99, 103, 107, 109, 111, 118, 122, 123, 125, 127, 129, 130, 131, 133, 134, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A046051, A054992, A309942, A337811, A337812, A337813.

PARI
```
for(n=1,150,if(bigomega(2^n-1)
				
```

A337812 Numbers k such that the number of prime factors, counted with multiplicity, of 2^k - 1 is greater than the corresponding count for 2^k + 1.

Original entry on oeis.org

4, 6, 8, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 43, 44, 48, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 79, 80, 84, 87, 88, 90, 92, 94, 96, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 119, 120, 124, 126, 128, 132, 136, 138, 140, 144, 146, 148, 151

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A046051, A054992, A309942, A337810, A337811, A337813.

Select[Range[152],PrimeOmega[2^#-1]>PrimeOmega[2^#+1]&] (* Harvey P. Dale, Dec 31 2022 *)

for(n=1,160,if(bigomega(2^n-1)>bigomega(2^n+1),print1(n,", ")))

A193295 Number of prime divisors (with multiplicity) of n^2 - 1.

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 3, 5, 3, 5, 2, 5, 3, 6, 3, 7, 2, 6, 3, 5, 3, 6, 3, 6, 5, 5, 4, 6, 2, 8, 3, 7, 4, 6, 3, 6, 3, 6, 3, 7, 2, 6, 4, 5, 4, 7, 3, 8, 4, 6, 3, 7, 3, 8, 4, 6, 3, 6, 2, 6, 4, 8, 5, 9, 3, 6, 3, 6, 3, 8, 2, 7, 4, 5, 5, 6, 3, 8, 5, 7, 5, 6, 3, 6, 4, 6

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 22 2011

nonn

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A046051, A054992, A054990, A054991, A082863.

Table[PrimeOmega[n^2 - 1], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(n^2-1) \\ Charles R Greathouse IV, Jul 30 2011

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

A226116 Numbers k such that one of 2^k-1 or 2^k+1 is semiprime, but not both.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 12, 13, 17, 19, 20, 28, 31, 32, 37, 40, 41, 43, 49, 59, 61, 64, 67, 79, 83, 92, 97, 103, 104, 109, 127, 128, 131, 137, 139, 148, 149, 191, 197, 227, 241, 256, 269, 271, 281, 293, 313, 356, 373, 379, 421, 457, 487, 523, 596, 692, 701, 727, 809, 881, 971, 983, 997, 1004, 1061, 1063

Offset: 1

Irina Gerasimova, May 28 2013

			2^3-1=7 is not a semiprime but 2^3+1 =9 is, so 3 is in the sequence.
2^4-1 =15 is a semiprime but 2^4+1 =17 is not, so 4 is in the sequence.
2^8-1 =255 is a 3-prime (not a 2-prime) and 2^8+1 =257 is a prime (not a 2-prime), so 8 is not in the sequence.

Cf. A001358, A046051, A054992, A092558, A092559.

isok(n) = {nbm = bigomega(2^n-1); nbp = bigomega(2^n+1); return (((nbm == 2) || (nbp == 2)) && ! ((nbm == 2) && (nbp == 2)));} \\ Michel Marcus, Aug 23 2013

Original sequence of 4 small numbers replaced by a wider sequence. - R. J. Mathar, Jun 13 2013

A226368 Numbers k such that Omega(k) = Omega(2^k + 1), where Omega = A001222 is the number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 6, 36, 44, 52, 60, 72, 88, 112, 116, 136, 140, 152, 184, 288, 292, 320, 352, 388, 400, 404, 536, 544, 584, 632, 796, 844, 928, 1072, 1136

Offset: 1

Irina Gerasimova, Jun 05 2013

At the moment, the least candidate for a(31) is 1168 = 2^4 * 73; the entry for 2^1168 + 1 on factordb.com has 3 prime factors and 1 composite cofactor with 326 decimal digits. - Lucas A. Brown, Mar 19 2024

Cf. A001222, A054992, A155900.

is(n)=bigomega(2^n+1)==bigomega(n) \\ Charles R Greathouse IV, Mar 18 2014

{k: A054992(k) = A001222(k)}.

More terms from Jon E. Schoenfield, Sep 01 2013
a(29) added using factordb.com by Daniel Suteu, Jan 21 2023
a(30) added using factordb.com by Lucas A. Brown, Mar 19 2024

cp's OEIS Frontend

A366687 Number of prime factors of 11^n + 1 (counted with multiplicity).

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A001269 Table T(n,k) in which n-th row lists prime factors of 2^n + 1 (n >= 0), with repetition.

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A337810 Numbers k such that the number of prime factors, counted with multiplicity, of 2^k - 1 is less than the corresponding count for 2^k + 1.

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A337812 Numbers k such that the number of prime factors, counted with multiplicity, of 2^k - 1 is greater than the corresponding count for 2^k + 1.

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Mathematica

PARI

A193295 Number of prime divisors (with multiplicity) of n^2 - 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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Magma

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A226116 Numbers k such that one of 2^k-1 or 2^k+1 is semiprime, but not both.

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PARI

Extensions

A226368 Numbers k such that Omega(k) = Omega(2^k + 1), where Omega = A001222 is the number of prime factors counted with multiplicity.

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