cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A057954 Number of prime factors of 7^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 5, 4, 8, 3, 8, 4, 10, 7, 8, 4, 13, 3, 9, 7, 13, 4, 12, 4, 14, 7, 9, 5, 18, 5, 8, 12, 13, 5, 14, 5, 16, 9, 8, 7, 18, 5, 9, 8, 18, 5, 15, 4, 15, 12, 9, 4, 22, 8, 11, 10, 13, 5, 18, 6, 19, 10, 9, 6, 24, 6, 11, 11, 20, 9, 17, 6, 14, 10, 18, 4, 26, 7, 10, 11, 13, 9, 17, 4, 24, 17, 12, 9, 22
Offset: 1

Views

Author

Patrick De Geest, Nov 15 2000

Keywords

Examples

			7^8-1 = 5764800 = 2^6 * 3 * 5^2 * 1201 and a(8) = bigomega(5764800) = 6+1+2+1 = 10. - _Bernard Schott_, Feb 02 2020

Links

Crossrefs

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), this sequence (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A024075, A059889, A074249, A085032.

Programs

  • Magma
    f:=func; [f(7^n- 1):n in [1..85]]; // Marius A. Burtea, Feb 02 2020
  • Mathematica
    PrimeOmega[Table[7^n - 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)

Formula

Möbius transform of A085032. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024075(n)). - Amiram Eldar, Feb 02 2020

A059889 a(n) = |{m : multiplicative order of 7 mod m=n}|.

Original entry on oeis.org

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
Offset: 1

Views

Author

Vladeta Jovovic, Feb 06 2001

Keywords

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

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.

Programs

  • 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
  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#] * DivisorSigma[0, 7^#-1] &]; Array[a, 60] (* Amiram Eldar, Jan 25 2025 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d) * numdiv(7^d-1)); \\ Amiram Eldar, Jan 25 2025

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).

A274906 Largest prime factor of 4^n - 1.

Original entry on oeis.org

3, 5, 7, 17, 31, 13, 127, 257, 73, 41, 683, 241, 8191, 127, 331, 65537, 131071, 109, 524287, 61681, 5419, 2113, 2796203, 673, 4051, 8191, 262657, 15790321, 3033169, 1321, 2147483647, 6700417, 599479, 131071, 122921, 38737, 616318177, 525313, 22366891
Offset: 1

Views

Author

Vincenzo Librandi, Jul 11 2016

Keywords

Examples

			4^7 - 1 = 16383 = 3*43*127, so a(7) = 127

Links

Crossrefs

Second bisection of A005420. - Michel Marcus, Jul 13 2016
Cf. largest prime factor of k^n-1: A005420 (k=2), A074477 (k=3), this sequence (k=4), A074479 (k=5), A274907 (k=6), A074249 (k=7), A274908 (k=8), A274909 (k=9), A005422 (k=10), A274910 (k=11).
Cf. A006530, A024036.

Programs

  • Magma
    [Maximum(PrimeDivisors(4^n-1)): n in [1..40]];
  • Mathematica
    Table[FactorInteger[4^n - 1][[-1, 1]], {n, 40}]

Formula

a(n) = A006530(A024036(n)). - Michel Marcus, Jul 11 2016
a(n) = max(A002587(n),A005420(n)). - Max Alekseyev, Apr 25 2022

Extensions

Terms to a(100) in b-file from Vincenzo Librandi, Jul 13 2016
a(101)-a(603) in b-file from Amiram Eldar, Feb 08 2020
a(604)-a(1128) in b-file from Max Alekseyev, Jul 25 2023, Mar 15 2025

A074479 Largest prime factor of 5^n - 1.

Original entry on oeis.org

2, 3, 31, 13, 71, 31, 19531, 313, 829, 521, 12207031, 601, 305175781, 19531, 1741, 11489, 466344409, 5167, 3981071, 9161, 519499, 12207031, 332207361361, 390001, 9384251, 305175781, 31051, 234750601, 22125996444329, 7621
Offset: 1

Views

Author

Rick L. Shepherd, Aug 23 2002

Keywords

Examples

			5^9 - 1 = 1953124 = (2^2)*19*31*829, so a(9) = 829.

Links

Crossrefs

Cf. A074478 (largest prime factor of 5^n + 1), A074477 (largest prime factor of 3^n - 1), A074249 (largest prime factor of 7^n - 1).
Cf. A006530, A024049.
Cf. similar sequences listed in A274906.

Programs

  • Magma
    [Maximum(PrimeDivisors(5^n-1)): n in [1..45]]; // Vincenzo Librandi, Jul 13 2016
  • Mathematica
    Table[FactorInteger[5^n - 1] [[-1, 1]], {n, 30}] (* Vincenzo Librandi, Aug 23 2013 *)
  • PARI
    for(n=1,32, v=factor(5^n-1); print1(v[matsize(v)[1],1],","))

Formula

a(n) = A006530(A024049(n)). - Vincenzo Librandi, Jul 13 2016

Extensions

Terms to a(100) in b-file from Vincenzo Librandi, Aug 23 2013
a(101)-a(448) in b-file from Amiram Eldar, Feb 01 2020
a(449)-a(502) in b-file from Max Alekseyev, Apr 25 2022

A379640 Smallest primitive prime factor of 7^n-1.

Original entry on oeis.org

2, 1, 19, 5, 2801, 43, 29, 1201, 37, 11, 1123, 13, 16148168401, 113, 31, 17, 14009, 117307, 419, 281, 11898664849, 23, 47, 73, 2551, 53, 109, 13564461457, 59, 6568801, 311, 353, 3631, 29078814248401, 2127431041, 13841169553, 223, 351121, 486643, 41, 83, 51031
Offset: 1

Views

Author

Sean A. Irvine, Dec 28 2024

Keywords

Comments

Also, smallest prime p such that 1/p has septimal period n.

Links

Crossrefs

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).
Cf. A074249.

Programs

  • PARI
    listap(nn) = {prf = []; for (n=1, nn, vp = (factor(7^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

A085032 Number of prime factors of cyclotomic(n,7), which is A019325(n), the value of the n-th cyclotomic polynomial evaluated at x=7.

Original entry on oeis.org

2, 3, 2, 3, 1, 1, 2, 2, 3, 2, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 1, 2, 3, 3, 2, 2, 5, 1, 3, 1, 3, 3, 3, 1, 2, 1, 3, 2, 3, 2, 3, 2, 2, 3, 2, 1, 2, 1, 4, 1, 4, 2, 3, 1, 1, 4, 4, 1, 4, 2, 4, 3, 1, 4, 5, 2, 4, 3, 3, 4, 2, 3, 5, 2, 2, 1, 3, 3, 2, 3, 5, 4, 7, 1
Offset: 1

Views

Author

T. D. Noe, Jun 19 2003

Keywords

Comments

The Mobius transform of this sequence yields A057954, number of prime factors of 7^n-1.

References

Links

Crossrefs

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), this sequence (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).
Cf. A019325, A057954, A059889, A074249.

Programs

  • Mathematica
    Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 7]]][[2]], {n, 1, 100}]

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

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A024075, A001221, A057954, A059889, A074249, A218358, A366620, A366633, A366634, A366635.
Cf. A046800, A133801, A366604, A366611, A366620, A366651, A366660, A102347, A366681, A366707.

Programs

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

Formula

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

A366633 Number of divisors of 7^n-1.

Original entry on oeis.org

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
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(5)=8 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 168061}.

Links

Crossrefs

Cf. A024075, A000005, A057954, A059889, A074249, A218358, A366632, A366634, A366635.
Cf. A046801, A366575, A366602, A366612, A366621, A366652, A366661, A070528, A366683, A366709.

Programs

  • Maple
    a:=n->numtheory[tau](7^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[0, 7^Range[100]-1]
  • PARI
    a(n) = numdiv(7^n-1);

Formula

a(n) = sigma0(7^n-1) = A000005(A024075(n)).

A366634 Sum of the divisors of 7^n-1.

Original entry on oeis.org

12, 124, 780, 7812, 33624, 354640, 1704240, 18929096, 97036800, 800520192, 3958188480, 56928231360, 193778020824, 1830926384640, 11181115146240, 115997032277280, 465294239722800, 5175558387507200, 22852200371636160, 287850454432579584, 1318081737957660000
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(5)=33624 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 16806}.

Links

Crossrefs

Cf. A024075, A000203, A057954, A059887, A074249, A218358, A366632, A366633, A366635.
Cf. A075708, A366576, A366603, A366613, A366622, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](7^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1, 7^Range[30]-1]

Formula

a(n) = sigma(7^n-1) = A000203(A024075(n)).

A227575 Largest prime factor of 7^n + 1.

Original entry on oeis.org

2, 2, 5, 43, 1201, 191, 181, 911, 169553, 117307, 4021, 10746341, 1201, 228511817, 13564461457, 6568801, 47072139617, 29078814248401, 13841169553, 4058036683, 810221830361, 309079, 83960385389, 3421093417510114543, 33232924804801, 79787519018560501
Offset: 0

Views

Author

Michel Marcus, Aug 22 2013

Keywords

Examples

			7^12 + 1 = 2*73*193*409*1201, so a(12) = 1201.

Links

Crossrefs

Cf. A006530, A034491.
Cf. A002587, A074249, A074476, A074478.

Programs

  • Magma
    [Maximum(PrimeDivisors(7^n+1)): n in [0..30]]; // Bruno Berselli, Aug 23 2013
  • Mathematica
    Table[FactorInteger[7^n + 1][[-1, 1]], {n, 0, 30}] (* Bruno Berselli, Aug 23 2013 *)
  • PARI
    a(n) = f = factor(7^n + 1); f[#f~, 1]; \\ Michel Marcus, Aug 22 2013

Formula

a(n) = A006530(A034491(n)). - Vincenzo Librandi, Jul 12 2016

Extensions

Terms to a(100) in b-file from Vincenzo Librandi, Jul 12 2016
a(101)-a(372) in b-file from Amiram Eldar, Feb 02 2020
a(373)-a(387) in b-file from Max Alekseyev, Apr 25 2022, Aug 30 2023
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.