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Search: a057954 -id:a057954
Displaying 1-10 of 18 results found. page 1 2
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A046051 Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity). +10
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
A057951 Number of prime factors of 10^n - 1 (counted with multiplicity). +10
26
2, 3, 4, 4, 4, 7, 4, 6, 6, 6, 4, 9, 5, 6, 8, 8, 4, 11, 3, 9, 9, 9, 3, 12, 7, 8, 9, 10, 7, 15, 5, 13, 8, 8, 9, 14, 5, 5, 8, 13, 6, 17, 6, 13, 12, 8, 4, 15, 6, 12, 10, 11, 6, 16, 10, 14, 8, 10, 4, 22, 9, 7, 16, 17, 9, 17, 5, 12, 8, 14, 4, 20, 5, 9, 14, 8, 10, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of 11...11 (Repunit).
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085035 - T. D. Noe, Jun 19 2003
a(n) = Omega(10^n -1) = Omega(R_n) + 2 = A046053(n) + 2 {where Omega(n) = A001222(n) and R_n = (10^n - 1)/9 = A002275(n)}. - Lekraj Beedassy, Jun 09 2006
a(n) = A001222(A002283(n)). - Ray Chandler, Apr 22 2017
MATHEMATICA
PrimeOmega[10^Range[70]-1] (* Jayanta Basu, May 29 2013 *)
PROG
(PARI) a(n)=bigomega(10^n-1) \\ Charles R Greathouse IV, Sep 14 2015
CROSSREFS
bigomega(b^n-1): this sequence (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A002283, A046053, A085035, A003020, A046107, A005422, A061075, A102380, A095370, A147556, A081317, A081318, A102347, A112505.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
EXTENSIONS
Erroneous b-file replaced by Ray Chandler, Apr 26 2017
STATUS
approved
A057958 Number of prime factors of 3^n - 1 (counted with multiplicity). +10
22
1, 3, 2, 5, 3, 5, 2, 7, 3, 6, 3, 8, 2, 5, 5, 10, 3, 8, 3, 10, 4, 7, 3, 11, 5, 5, 6, 9, 4, 11, 4, 12, 5, 8, 6, 12, 3, 7, 7, 13, 4, 11, 3, 11, 9, 6, 5, 17, 7, 10, 6, 9, 4, 13, 8, 13, 7, 9, 3, 17, 3, 8, 6, 14, 7, 12, 4, 12, 6, 11, 2, 16, 5, 8, 10, 11, 7, 15, 4, 18, 9, 8, 5, 18, 7, 6, 8, 16, 4, 19, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..690 (first 660 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085028. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024023(n)). - Amiram Eldar, Feb 01 2020
PROG
(PARI) a(n)=bigomega(3^n-1) \\ Charles R Greathouse IV, Sep 14 2015
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), this sequence (b=3), A046051 (b=2).
Cf. A001222, A024023, A085028, A002591 A074477, A057952, A133801, A274909.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
EXTENSIONS
Offset corrected by Amiram Eldar, Feb 01 2020
STATUS
approved
A057957 Number of prime factors of 4^n - 1 (counted with multiplicity). +10
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
A057953 Number of prime factors of 8^n - 1 (counted with multiplicity). +10
18
1, 3, 2, 5, 3, 6, 4, 7, 3, 7, 4, 10, 4, 8, 6, 10, 5, 9, 4, 13, 7, 9, 4, 14, 7, 8, 6, 14, 6, 13, 3, 13, 8, 11, 11, 15, 6, 9, 9, 17, 5, 14, 5, 15, 10, 9, 6, 19, 7, 14, 8, 18, 8, 16, 10, 19, 7, 11, 6, 24, 5, 8, 10, 16, 8, 17, 6, 20, 9, 22, 7, 21, 7, 13, 14, 17, 10, 16, 8, 23, 10, 12, 5, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..500 (first 402 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085033. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024088(n)) = A046051(3*n). - Amiram Eldar, Feb 02 2020
MATHEMATICA
PrimeOmega/@(8^Range[90]-1) (* Harvey P. Dale, May 24 2018 *)
PROG
(Magma) f:=func<n|&+[p[2]: p in Factorization(n)]>; [f(8^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020
CROSSREFS
bigomega(b^n-1): A057951 (b=10), A057952 (b=9), this sequence (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A024088, A085033, A274908.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
A057952 Number of prime factors of 9^n - 1 (counted with multiplicity). +10
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
A057955 Number of prime factors of 6^n - 1 (counted with multiplicity). +10
17
1, 2, 2, 3, 3, 4, 2, 4, 4, 6, 3, 7, 3, 6, 6, 6, 5, 7, 3, 9, 4, 5, 5, 9, 6, 7, 6, 9, 2, 11, 3, 9, 6, 8, 7, 13, 6, 6, 6, 12, 3, 10, 5, 9, 11, 8, 4, 13, 5, 10, 9, 11, 4, 11, 7, 14, 7, 6, 4, 20, 4, 5, 10, 12, 9, 12, 3, 11, 8, 18, 2, 18, 5, 10, 12, 9, 6, 15, 4, 17, 8, 7, 8, 17, 10, 7, 7, 12, 4, 18, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..420
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Möbius transform of A085031. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024062(n)). - Amiram Eldar, Feb 02 2020
EXAMPLE
6^10 - 1 = 60466175 = 5^2 * 7 * 11 * 101 * 311 and a(10) = bigomega(60466175) = 2+1+1+1+1 = 6. - Bernard Schott, Feb 02 2020
MATHEMATICA
PrimeOmega[6^Range[100]-1] (* Harvey P. Dale, Dec 14 2015 *)
PROG
(Magma) f:=func<n|&+[p[2]: p in Factorization(n)]>; [f(6^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020
CROSSREFS
bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), this sequence (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A024062, A085031, A274907.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
A057956 Number of prime factors of 5^n - 1 (counted with multiplicity). +10
17
2, 4, 3, 6, 4, 7, 3, 8, 5, 7, 3, 10, 3, 7, 7, 11, 4, 11, 5, 11, 6, 8, 4, 13, 8, 7, 9, 10, 5, 14, 4, 14, 6, 8, 9, 16, 5, 10, 6, 15, 4, 16, 4, 12, 12, 8, 3, 17, 4, 13, 8, 12, 5, 19, 10, 13, 7, 9, 4, 21, 5, 9, 11, 18, 8, 15, 7, 14, 9, 16, 4, 22, 5, 10, 16, 14, 7, 14, 5, 20, 11, 10, 5, 22, 9, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..502 (first 448 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085030. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024049(n)). - Amiram Eldar, Feb 01 2020
MATHEMATICA
PrimeOmega[5^Range[90]-1] (* Harvey P. Dale, Dec 16 2017 *)
CROSSREFS
bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), this sequence (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A024049, A074479, A085030.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
A059889 a(n) = |{m : multiplicative order of 7 mod m=n}|. +10
16
4, 6, 8, 26, 4, 42, 12, 48, 52, 66, 12, 778, 4, 138, 80, 300, 12, 528, 12, 1430, 72, 138, 28, 15216, 24, 66, 1216, 966, 28, 3630, 28, 1344, 360, 58, 108, 16988, 28, 138, 176, 12752, 28, 7398, 12, 4422, 1900, 122, 12, 131028, 240, 536, 744, 1046, 28, 23744, 44 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
a(n) = number of orders of degree n monic irreducible polynomials over GF(7).
Also, number of primitive factors of 7^n - 1 (cf. A218358). - Max Alekseyev, May 03 2022
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..388
FORMULA
a(n) = Sum_{d|n} mu(n/d)*tau(7^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).
MAPLE
with(numtheory):
a:= n-> add(mobius(n/d)*tau(7^d-1), d=divisors(n)):
seq(a(n), n=1..40); # Alois P. Heinz, Oct 12 2012
CROSSREFS
Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), this sequence (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A058946, A053450, A057954, A058946, A074249, A212486, A218358
Column k=7 of A212957.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Feb 06 2001
STATUS
approved
A057937 Number of prime factors of 7^n + 1 (counted with multiplicity). +10
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
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Last modified August 29 18:55 EDT 2024. Contains 375518 sequences. (Running on oeis4.)