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Search: a057954 -id:a057954
Displaying 1-10 of 18 results found.
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A046051
Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).
+0
45
0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, 1, 6, 4, 4, 2, 7, 3, 3, 3, 6, 3, 7, 1, 5, 4, 3, 4, 10, 2, 3, 4, 8, 2, 8, 3, 7, 6, 4, 3, 10, 2, 7, 5, 7, 3, 9, 6, 8, 4, 6, 2, 13, 1, 3, 7, 7, 3, 9, 2, 7, 4, 9, 3, 14, 3, 5, 7, 7, 4, 8, 3, 10, 6, 5, 2, 14, 3, 5, 6, 10, 1, 13, 5, 9, 3, 6, 5, 13, 2, 5, 8
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Length of row n of A001265.
LINKS
Sean A. Irvine, Table of n, a(n) for n = 1..1206 (terms 1..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.
Alex Kontorovich, Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number
FORMULA
Mobius transform of A085021. - T. D. Noe, Jun 19 2003
a(n) = A001222(A000225(n)). - Michel Marcus, Jun 06 2019
EXAMPLE
a(4) = 2 because 2^4 - 1 = 15 = 3*5.
From Gus Wiseman, Jul 04 2019: (Start)
The sequence of Mersenne numbers together with their prime indices begins:
1: {}
3: {2}
7: {4}
15: {2,3}
31: {11}
63: {2,2,4}
127: {31}
255: {2,3,7}
511: {4,21}
1023: {2,5,11}
2047: {9,24}
4095: {2,2,3,4,6}
8191: {1028}
16383: {2,14,31}
32767: {4,11,36}
65535: {2,3,7,55}
131071: {12251}
262143: {2,2,2,4,8,21}
524287: {43390}
1048575: {2,3,3,5,11,13}
(End)
MATHEMATICA
a[q_] := Module[{x, n}, x=FactorInteger[2^n-1]; n=Length[x]; Sum[Table[x[i][2], {i, n}][j], {j, n}]]
a[n_Integer] := PrimeOmega[2^n - 1]; Table[a[n], {n, 200}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
PROG
(PARI) a(n)=bigomega(2^n-1) \\ Charles R Greathouse IV, Apr 01 2013
CROSSREFS
bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), this sequence (b=2).
Cf. A000043, A000668, A001348, A054988, A054989, A054990, A054991, A054992, A085021.
Cf. A000225, A001221, A001222, A046800, A049093, A059305, A325610, A325611, A325612, A325625.
KEYWORD
nonn
AUTHOR
Eric W. Weisstein
STATUS
approved
A366634
Sum of the divisors of 7^n-1.
+0
11
12, 124, 780, 7812, 33624, 354640, 1704240, 18929096, 97036800, 800520192, 3958188480, 56928231360, 193778020824, 1830926384640, 11181115146240, 115997032277280, 465294239722800, 5175558387507200, 22852200371636160, 287850454432579584, 1318081737957660000
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..388
FORMULA
a(n) = sigma(7^n-1) = A000203(A024075(n)).
EXAMPLE
a(5)=33624 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 16806}.
MAPLE
a:=n->numtheory[sigma](7^n-1):
seq(a(n), n=1..100);
MATHEMATICA
DivisorSigma[1, 7^Range[30]-1]
CROSSREFS
Cf. A024075, A000203, A057954, A059887, A074249, A218358, A366632, A366633, A366635.
Cf. A075708, A366576, A366603, A366613, A366622, A366653, A366662, A102146, A366684, A366710.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 14 2023
STATUS
approved
A366633
Number of divisors of 7^n-1.
+0
11
4, 10, 12, 36, 8, 60, 16, 84, 64, 80, 16, 864, 8, 160, 96, 384, 16, 640, 16, 1536, 96, 160, 32, 16128, 32, 80, 1280, 1152, 32, 3840, 32, 1728, 384, 80, 128, 18432, 32, 160, 192, 14336, 32, 7680, 16, 4608, 2048, 160, 16, 147456, 256, 640, 768, 1152, 32, 25600
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..388
FORMULA
a(n) = sigma0(7^n-1) = A000005(A024075(n)).
EXAMPLE
a(5)=8 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 168061}.
MAPLE
a:=n->numtheory[tau](7^n-1):
seq(a(n), n=1..100);
MATHEMATICA
DivisorSigma[0, 7^Range[100]-1]
PROG
(PARI) a(n) = numdiv(7^n-1);
CROSSREFS
Cf. A024075, A000005, A057954, A059889, A074249, A218358, A366632, A366634, A366635.
Cf. A046801, A366575, A366602, A366612, A366621, A366652, A366661, A070528, A366683, A366709.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 14 2023
STATUS
approved
A366632
Number of distinct prime divisors of 7^n - 1.
+0
11
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..388
FORMULA
a(n) = omega(7^n-1) = A001221(A024075(n)).
PROG
(PARI) for(n = 1, 100, print1(omega(7^n - 1), ", "))
CROSSREFS
Cf. A024075, A001221, A057954, A059889, A074249, A218358, A366620, A366633, A366634, A366635.
Cf. A046800, A133801, A366604, A366611, A366620, A366651, A366660, A102347, A366681, A366707.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 14 2023
STATUS
approved
A057952
Number of prime factors of 9^n - 1 (counted with multiplicity).
+0
17
3, 5, 5, 7, 6, 8, 5, 10, 8, 10, 7, 11, 5, 9, 11, 12, 8, 12, 7, 13, 11, 11, 6, 17, 10, 9, 13, 13, 9, 17, 8, 14, 12, 12, 11, 16, 8, 11, 15, 18, 8, 18, 6, 16, 19, 10, 10, 21, 12, 18, 15, 13, 8, 18, 15, 19, 15, 13, 7, 24, 7, 13, 19, 16, 12, 18, 8, 17, 15, 20, 9, 24, 9, 13, 22, 17, 13, 22
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..690 (first 330 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085034. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024101(n)) = A057958(2*n). - Amiram Eldar, Feb 02 2020
a(n) = A057941(n) + A057958(n). - Max Alekseyev, Jan 07 2024
MATHEMATICA
PrimeOmega[Table[9^n - 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)
CROSSREFS
bigomega(b^n-1): A046051 (b=2), A057958 (b=3), A057957 (b=4), A057956 (b=5), A057955 (b=6), A057954 (b=7), A057953 (b=8), this sequence (b=9), A057951 (b=10), A366682 (b=11), A366708 (b=12).
Cf. A001222, A002591, A024101, A074477, A085034, A133801, A274909, A366660, A366661, A366662.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
A366682
Number of prime factors of 11^n - 1 (counted with multiplicity).
+0
7
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..316
FORMULA
a(n) = bigomega(11^n-1) = A001222(A024127(n)).
MATHEMATICA
PrimeOmega[11^Range[70]-1]
PROG
(PARI) a(n)=bigomega(11^n-1)
CROSSREFS
Cf. A024127, A001222, A366681, A366683, A366684, A366685, A366687.
Cf. A046051, A057958, A057957, A057956, A057955, A057954, A057953, A057952, A057951, A366708.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 16 2023
STATUS
approved
A366708
Number of prime factors of 12^n - 1 (counted with multiplicity).
+0
9
1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 4, 8, 3, 6, 6, 9, 3, 9, 2, 8, 5, 6, 4, 12, 4, 8, 6, 10, 5, 13, 5, 11, 8, 6, 9, 14, 3, 6, 7, 14, 4, 14, 5, 12, 12, 8, 3, 18, 5, 10, 6, 13, 7, 16, 8, 13, 7, 8, 4, 19, 4, 8, 8, 13, 8, 17, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 11, 18, 4
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..310
FORMULA
a(n) = bigomega(12^n-1) = A001222(A024140(n)).
MATHEMATICA
PrimeOmega[12^Range[70]-1]
PROG
(PARI) a(n)=bigomega(12^n-1)
CROSSREFS
Cf. A024140, A001222, A046051, A057951, A057952, A057953, A057954, A057955, A057956, A057957, A057958, A250288, A252170, A366707, A366709, A366682.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 17 2023
STATUS
approved
A218358
Minimal order of degree-n irreducible polynomials over GF(7).
+0
8
1, 4, 9, 5, 2801, 36, 29, 64, 27, 11, 1123, 13, 16148168401, 113, 31, 17, 14009, 108, 419, 55, 261, 23, 47, 73, 2551, 53, 81, 145, 59, 99, 311, 256, 3631, 56036, 81229, 135, 223, 1676, 486643, 41, 83, 1017, 166003607842448777, 115, 837, 188, 13722816749522711
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) < 7^n.
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..430
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order
FORMULA
a(n) = min(M(n)) with M(n) = {d : d|(7^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.
a(n) = A212486(n,1) = A213224(n,4).
MAPLE
with(numtheory):
M:= proc(n) M(n):= divisors(7^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..42);
MATHEMATICA
M[n_] := M[n] = Divisors[7^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 47}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)
CROSSREFS
Cf. A057954, A074249, A212486, A213224.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 27 2012
STATUS
approved
A057937
Number of prime factors of 7^n + 1 (counted with multiplicity).
+0
15
3, 3, 4, 2, 5, 5, 5, 3, 5, 6, 5, 5, 5, 4, 7, 3, 4, 6, 5, 4, 8, 6, 4, 4, 6, 5, 6, 6, 4, 10, 6, 4, 8, 6, 11, 8, 5, 4, 9, 6, 7, 7, 10, 5, 12, 7, 4, 6, 12, 11, 8, 5, 7, 9, 13, 8, 9, 7, 7, 9, 4, 6, 14, 4, 9, 12, 6, 5, 8, 10, 5, 9, 6, 4, 11, 7, 12, 10, 7, 6, 10, 6, 5, 12, 8, 6, 8, 5, 8, 14, 11, 6, 9, 6, 11
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..387 (first 372 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
a(n) = A057954(2n) - A057954(n). - T. D. Noe, Jun 19 2003
a(n) = A001222(A034491(n)). - Amiram Eldar, Feb 02 2020
MATHEMATICA
PrimeOmega[Table[7^n + 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)
PROG
(Magma) f:=func<n|&+[p[2]: p in Factorization(n)]>; [f(7^n + 1):n in [1..110]]; // Marius A. Burtea, Feb 02 2020
CROSSREFS
bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), this sequence (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).
Cf. A001222, A034491, A057954, A227575.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Oct 15 2000
STATUS
approved
A057957
Number of prime factors of 4^n - 1 (counted with multiplicity).
+0
19
1, 2, 3, 3, 3, 5, 3, 4, 6, 6, 4, 7, 3, 6, 7, 5, 3, 10, 3, 8, 8, 7, 4, 10, 7, 7, 9, 8, 6, 13, 3, 7, 9, 7, 9, 14, 5, 7, 8, 10, 5, 14, 5, 10, 13, 9, 6, 13, 5, 14, 11, 10, 6, 15, 12, 11, 9, 9, 6, 17, 3, 8, 14, 9, 9, 15, 5, 11, 9, 16, 6, 19, 6, 10, 14, 11, 10, 18, 5, 13, 16, 10, 8, 19, 7, 10, 11
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..1122 (first 603 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085029. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024036(n)) = A046051(2*n). - Amiram Eldar, Feb 01 2020
MATHEMATICA
PrimeOmega/@(4^Range[90]-1) (* Harvey P. Dale, Dec 31 2018 *)
CROSSREFS
bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), this sequence (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A024036, A054992, A085029, A274906.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
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