A046800 -id:A046800 - cp's OEIS frontend

A366632 Number of distinct prime divisors of 7^n - 1.

2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 4, 7, 3, 6, 6, 6, 4, 7, 4, 8, 6, 6, 5, 11, 5, 5, 9, 8, 5, 10, 5, 8, 8, 5, 7, 11, 5, 6, 7, 11, 5, 11, 4, 10, 10, 6, 4, 14, 8, 8, 9, 8, 5, 12, 6, 13, 8, 6, 6, 17, 6, 8, 9, 11, 9, 13, 6, 9, 9, 15, 4, 18, 7, 7, 10, 8, 9, 13, 4, 16, 13

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..388

Cf. A024075, A001221, A057954, A059889, A074249, A218358, A366620, A366633, A366634, A366635.

Cf. A046800, A133801, A366604, A366611, A366620, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(7^n - 1), ", "))

a(n) = omega(7^n-1) = A001221(A024075(n)).

A366660 Number of distinct prime divisors of 9^n - 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 3, 5, 6, 5, 5, 7, 3, 6, 8, 6, 6, 9, 5, 7, 8, 8, 4, 12, 7, 6, 11, 9, 7, 12, 6, 7, 10, 9, 8, 12, 6, 8, 12, 11, 6, 14, 4, 12, 16, 7, 8, 15, 10, 12, 13, 9, 6, 15, 11, 14, 13, 10, 5, 18, 5, 10, 16, 8, 9, 15, 6, 13, 13, 15, 7, 19, 7, 10, 19, 13, 11

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024101, A001221, A057952, A366661, A366662, A366663.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366651, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(9^n - 1), ", "))

a(n) = omega(9^n-1) = A001221(A024101(n)).

a(n) = A133801(2*n) = A133801(n) + A366580(n) - 1. - Max Alekseyev, Jan 07 2024

A060443 Table T(n,k) in which n-th row lists prime factors of 2^n - 1 (n >= 2), without repetition.

Original entry on oeis.org

0, 1, 3, 7, 3, 5, 31, 3, 7, 127, 3, 5, 17, 7, 73, 3, 11, 31, 23, 89, 3, 5, 7, 13, 8191, 3, 43, 127, 7, 31, 151, 3, 5, 17, 257, 131071, 3, 7, 19, 73, 524287, 3, 5, 11, 31, 41, 7, 127, 337, 3, 23, 89, 683, 47, 178481, 3, 5, 7, 13, 17, 241

Offset: 0

N. J. A. Sloane

For n > 1, the length of row n is A046800(n). - T. D. Noe, Aug 06 2007

			From _Wolfdieter Lang_, Sep 23 2017: (Start)
The irregular triangle T(n,k) begins for n >= 2:
n\k      1   2    3    4   5
2:       3
3:       7
4:       3   5
5:      31
6:       3   7
7:     127
8:       3   5   17
9:       7  73
10:      3  11   31
11:     23  89
12:      3   5    7   13
13:   8191
14:      3  43  127
15:      7  31  151
16:      3   5   17  257
17: 131071
18:      3   7   19   73
19: 524287
20:      3   5   11   31  41
... (End)

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

T. D. Noe, Rows n=0..500 of triangle, flattened (derived from Brillhart et al.)
Joerg Arndt, Rows n=1..1200 of triangle when repetitions are included (derived from Brillhart et al.)
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 [Broken link]
S. S. Wagstaff, Jr., The Cunningham Project
Chai Wah Wu, Tables from the Cunningham Project in machine-readable JSON format.

Cf. A001265, A064078.

Array[FactorInteger[2^# - 1][[All, 1]] &, 25, 0] (* Paolo Xausa, Apr 18 2024 *)

A366604 Number of distinct prime divisors of 4^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 3, 6, 6, 5, 3, 8, 3, 7, 6, 7, 4, 9, 7, 7, 6, 8, 6, 11, 3, 7, 8, 7, 9, 12, 5, 7, 7, 9, 5, 12, 5, 10, 11, 9, 6, 12, 5, 12, 10, 10, 6, 12, 11, 11, 8, 9, 6, 15, 3, 8, 11, 9, 9, 14, 5, 10, 8, 15, 6, 17, 6, 10, 13, 11, 10, 16, 5

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128

Cf. A024036, A001221, A057957, A133801, A366602, A366603.

Cf. A046800, A133801, A366611, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

PrimeNu[4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)

for(n = 1, 100, print1(omega(4^n - 1), ", "))

from sympy import primenu
def A366604(n): return primenu((1<<(n<<1))-1) # Chai Wah Wu, Oct 15 2023

a(n) = omega(4^n-1) = A001221(A024036(n)).

a(n) = A046800(2*n) = A046799(n) + A046800(n). - Max Alekseyev, Jan 07 2024

A366651 Number of distinct prime divisors of 8^n - 1.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 6, 3, 6, 4, 8, 4, 6, 6, 9, 5, 6, 4, 11, 6, 8, 4, 12, 7, 7, 6, 12, 6, 11, 3, 12, 8, 10, 10, 12, 6, 8, 9, 15, 5, 11, 5, 14, 10, 8, 6, 17, 5, 13, 8, 16, 8, 12, 10, 17, 7, 10, 6, 21, 5, 7, 9, 15, 8, 15, 6, 19, 9, 20, 7, 18, 7, 12, 14, 16, 9

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..500

Cf. A024088, A001221, A057953, A366652, A366653, A366654.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(8^n - 1), ", "))

a(n) = omega(8^n-1) = A001221(A024088(n)).

a(n) = A046800(3*n). - Max Alekseyev, Jan 09 2024

A366611 Number of distinct prime divisors of 5^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 5, 2, 6, 2, 5, 6, 6, 3, 7, 4, 8, 5, 6, 3, 8, 7, 5, 8, 7, 4, 11, 3, 8, 5, 6, 8, 11, 4, 8, 5, 11, 3, 12, 3, 9, 11, 6, 2, 11, 3, 11, 7, 8, 4, 14, 8, 9, 6, 7, 3, 17, 4, 7, 10, 11, 7, 12, 6, 11, 8, 14, 3, 16, 4, 8, 15, 11, 6, 11, 4, 15

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A001221, A057956, A059887, A074479, A143665, A218357, A295502, A366612, A366613.

Cf. A046800, A133801, A366604, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(5^n - 1), ", "))

a(n) = omega(5^n-1) = A001221(A024049(n)).

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

Original entry on oeis.org

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

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}]

cp's OEIS Frontend

A366632 Number of distinct prime divisors of 7^n - 1.

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A366660 Number of distinct prime divisors of 9^n - 1.

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A060443 Table T(n,k) in which n-th row lists prime factors of 2^n - 1 (n >= 2), without repetition.

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A366604 Number of distinct prime divisors of 4^n - 1.

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A366651 Number of distinct prime divisors of 8^n - 1.

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A366611 Number of distinct prime divisors of 5^n - 1.

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

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A325610 Adjusted frequency depth of 2^n - 1.

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

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