A046051 -id:A046051 - cp's OEIS frontend

A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

1, 1, 2, 2, 1, 4, 1, 4, 5, 3, 6, 7, 4, 5, 7, 6, 7, 11, 7, 7, 9, 10, 7, 13, 7, 11, 9, 11, 11, 13, 11, 12, 15, 16, 10, 19, 19, 15, 18, 16, 16, 18, 10, 18, 18, 17, 15, 21, 15, 18, 24, 23, 19, 23, 25, 25, 18, 26, 25, 28, 21, 21, 25, 23, 21, 29, 28, 31, 21, 24, 23

Offset: 1

Gus Wiseman, May 12 2019

nonn

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
For n > 1, a(n) is the multiplicity of q(1) = 2 in the q-factorization of 2^n - 1.

			The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 6 leaves (o's), so a(11) = 6.

Keith Briggs, Matula numbers and rooted trees.

Cf. A001222, A001221, A056239, A112798.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325625.

mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];
Table[mglv[2^n-1],{n,30}]

More terms from Jinyuan Wang, Feb 25 2025

A088863 Number of prime factors of n-th Mersenne number M(p_n).

Original entry on oeis.org

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

Offset: 1

Jeppe Stig Nielsen, Nov 25 2003

nonn

			a(5)=2 because M(p_5)=M(11)=2047 has 2 (not necessarily distinct) prime factors.

Max Alekseyev, Table of n, a(n) for n = 1..197 (first 137 terms from Herman Jamke)
Herman Jamke, The first 137 terms in detail

Cf. A001348, A003260, A046051, A046932.

seq(nops(ifactor(2^ithprime(n)-1)),n=1..32); # Emeric Deutsch, Dec 23 2004

Do[m = 2^Prime[n] - 1; Print[Plus @@ Last /@ FactorInteger[m]], {n, 1, 50}] (* Ryan Propper, Jul 31 2005 *)

for(n=1,137,print1(bigomega(2^prime(n)-1)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

a(n) = A001222(A001348(n)) = A001222(A000225(A000040(n))).

14 more terms from Emeric Deutsch, Dec 23 2004
More terms from Ryan Propper, Jul 31 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

A325610 Adjusted frequency depth of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 3, 3, 3, 3, 5, 1, 3, 3, 3, 1, 5, 1, 5, 5, 3, 3, 5, 3, 3, 3, 3, 3, 5, 1, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 1, 3, 5, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 5, 3, 3, 3

Offset: 1

Gus Wiseman, May 12 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

Cf. A001222, A001221, A056239, A071625, A112798, A323014, A325280.

Mersenne numbers: A046051, A046800, A059305, A325611, A325612, A325625.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[fdadj[2^n-1],{n,100}]

A325611 Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1.

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 7, 10, 10, 12, 12, 15, 12, 14, 16, 18, 14, 20, 16, 23, 20, 22, 22, 25, 25, 24, 23, 29, 26, 30, 27, 31, 33, 28, 32, 38, 36, 31, 36, 40, 37, 38, 33, 43, 44, 42, 39, 48, 39, 49, 45, 48, 43, 49, 49, 53, 47, 54, 47, 61

Offset: 1

Gus Wiseman, May 12 2019

nonn

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
Then a(n) is one plus the number of factors (counted with multiplicity) in the q-factorization of 2^n - 1.

			The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 12 nodes (o's plus brackets), so a(11) = 12.

Cf. A001222, A001221, A056239, A112798.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Mersenne numbers: A046051, A046800, A059305, A325610, A325612, A325625.

mgwt[n_]:=If[n==1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>mgwt[PrimePi[p]]*k]]];
Table[mgwt[2^n-1],{n,30}]

A325625 Sorted prime signature of 2^n - 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 13 2019

The sorted prime signature of n is row n of A124010.

			We have 2^126 - 1 = 3^3 * 7^2 * 19 * 43 * 73 * 127 * 337 * 5419 * 92737 * 649657 * 77158673929, so row n = 126 is {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3}.
Triangle begins:
  1
  1
  1
  1 1
  1
  1 2
  1
  1 1 1
  1 1
  1 1 1
  1 1
  1 1 1 2
  1
  1 1 1
  1 1 1
  1 1 1 1
  1
  1 1 1 3
  1
  1 1 1 1 2

Cf. A001222, A056239, A067255, A071625, A112798, A118914, A124010.

Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325612.

Table[Sort[Last/@FactorInteger[2^n-1]],{n,30}]

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 5, 4, 7, 4, 9, 4, 9, 5, 8, 4, 13, 4, 8, 7, 12, 3, 12, 3, 11, 10, 11, 5, 17, 8, 10, 6, 13, 4, 15, 5, 15, 9, 9, 8, 17, 6, 10, 12, 15, 9, 17, 4, 15, 9, 12, 5, 24, 7, 14, 9, 13, 6, 16, 10, 19, 8, 10, 5, 21, 5, 12, 16, 19, 8, 22, 6, 15, 10, 19, 7, 24, 3, 11, 15

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..316

Cf. A024127, A001222, A366681, A366683, A366684, A366685, A366687.

Cf. A046051, A057958, A057957, A057956, A057955, A057954, A057953, A057952, A057951, A366708.

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

a(n) = bigomega(11^n-1) = A001222(A024127(n)).

A001265 Table T(n,k) in which n-th row lists prime factors of 2^n - 1 (n >= 2), with repetition.

Original entry on oeis.org

3, 7, 3, 5, 31, 3, 3, 7, 127, 3, 5, 17, 7, 73, 3, 11, 31, 23, 89, 3, 3, 5, 7, 13, 8191, 3, 43, 127, 7, 31, 151, 3, 5, 17, 257, 131071, 3, 3, 3, 7, 19, 73, 524287, 3, 5, 5, 11, 31, 41, 7, 7, 127, 337, 3, 23, 89, 683, 47, 178481, 3, 3, 5, 7, 13, 17, 241

Offset: 2

N. J. A. Sloane

For n > 1, the length of row n is A046051(n). - T. D. Noe, Aug 06 2007

			Table begins:
 n=2: 3;
 n=3: 7;
 n=4: 3, 5;
 n=5: 31;
 n=6: 3, 3, 7;
 n=7: 127;
 n=8: 3, 5, 17;
  ...

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 = 2..1206, flattened (rows 2..500 from T. D. Noe)
Joerg Arndt, Rows n = 1..1200 of triangle (derived from Brillhart et al.; updated by Jon E. Schoenfield)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Jeroen Demeyer, Machine-readable Cunningham Tables
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number
Chai Wah Wu, Tables from the Cunningham Project in machine-readable JSON format.

Cf. A060443, A182590.

Array[Flatten[ConstantArray[#1, #2] & @@ # & /@ FactorInteger[2^# - 1]] &, 24] // Flatten (* Michael De Vlieger, Dec 04 2017 *)

row(n)= if (n==1, return ([0])); my(f = factor(2^n-1), v = []); for (i=1, #f~, for (j=1, f[i, 2], v = concat(v, f[i,j]))); v; \\ Michel Marcus, Dec 05 2017

Ambiguous rows 0 and 1 removed by Max Alekseyev, Jul 25 2023

A172290 Prime divisors of 2^1092-1, listed with multiplicities.

Original entry on oeis.org

3, 3, 5, 7, 7, 13, 13, 29, 43, 53, 79, 113, 127, 157, 313, 337, 547, 911, 1093, 1093, 1249, 1429, 1613, 2731, 3121, 4733, 5419, 8191, 14449, 21841, 121369, 224771, 503413, 1210483, 1948129, 22366891, 108749551, 112901153, 23140471537, 25829691707, 105310750819, 467811806281, 4093204977277417, 8861085190774909, 556338525912325157, 86977595801949844993, 275700717951546566946854497, 292653113147157205779127526827, 3194753987813988499397428643895659569

Offset: 1

Artur Jasinski, Jan 30 2010

Up to now only two primes p such that p^2 divide 2^(p-1)-1 are known (these two are Wieferich primes, see A001220).
The sequence is finite with A001222(2^1092-1) = 49 terms; A001221(2^1092-1) = 45. - Reinhard Zumkeller, May 14 2010
Terms appearing more than once (in fact twice) are 3, 7, 13, and 1093.

Reinhard Zumkeller, Table of n, a(n) for n = 1..49 (complete sequence)
Dario A. Alpern, Factorization using the Elliptic Curve Method.

Cf. A001220, A172291, A177855, A046051, A046800, A242715.

Missing terms a(34) and a(35) inserted by Reinhard Zumkeller, May 14 2010

Definition clarified and terms corrected by Joerg Arndt, Apr 25 2011

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

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)
				
```

cp's OEIS Frontend

A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A088863 Number of prime factors of n-th Mersenne number M(p_n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A325610 Adjusted frequency depth of 2^n - 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A325611 Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325625 Sorted prime signature of 2^n - 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A001265 Table T(n,k) in which n-th row lists prime factors of 2^n - 1 (n >= 2), with repetition.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A172290 Prime divisors of 2^1092-1, listed with multiplicities.

Views

Author

Keywords

Comments

Links