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 1-8 of 8 results.

A064078 Zsigmondy numbers for a = 2, b = 1: Zs(n, 2, 1) is the greatest divisor of 2^n - 1 (A000225) that is coprime to 2^m - 1 for all positive integers m < n.

Original entry on oeis.org

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681
Offset: 1

Views

Author

Jens Voß, Sep 04 2001

Keywords

Comments

By Zsigmondy's theorem, the n-th Zsigmondy number for bases a and b is not 1 except in the three cases (1) a = 2, b = 1, n = 1, (2) a = 2, b = 1, n = 6, (3) n = 2 and a + b is a power of 2.
Composite terms are the maximal overpseudoprimes to base 2 (see A141232) for which the multiplicative order of 2 mod a(n) equals n. - Vladimir Shevelev, Aug 26 2008
a(n) = 2^n - 1 if and only if either n = 1 or n is prime. - Vladimir Shevelev, Sep 30 2008
a(n) == 1 (mod n), 2^(a(n)-1) == 1 (mod a(n)), A002326((a(n)-1)/2) = n. - Thomas Ordowski, Oct 25 2017
If n is odd, then the prime factors of a(n) are congruent to {1,7} mod 8, that is, they have 2 has a quadratic residue, and are congruent to 1 mod 2n. If n is divisible by 8, then the prime factors of a(n) are congruent to 1 mod 16. - Jianing Song, Apr 13 2019
Named after the Austrian mathematician Karl Zsigmondy (1867-1925). - Amiram Eldar, Jun 20 2021

Examples

			a(4) = 5 because 2^4 - 1 = 15 and its divisors being 1, 3, 5, 15, only 1 and 5 are coprime to 2^2 - 1 = 3 and 2^3 - 1 = 7, and 5 is the greater of these.
a(5) = 31 because 2^5 - 1 = 31 is prime.
a(6) = 1 because 2^6 - 1 = 63 and its divisors being 1, 3, 7, 9, 21, 63, only 1 is coprime to all of 3, 7, 15, 31.

Links

Crossrefs

Cf. A000225, A064079, A064080, A064081, A064082, A064083.
Cf. A019320, A063982.

Programs

  • Mathematica
    Table[Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]], {n, 40}] (* Alonso del Arte, Mar 14 2013 *)
  • PARI
    a(n) = my(m = polcyclo(n, 2)); m/gcd(m,n); \\ Michel Marcus, Mar 07 2015

Formula

Denominator of Sum_{d|n} d*moebius(n/d)/(2^d-1). - Vladeta Jovovic, Apr 02 2004
a(n) = A019320(n)/gcd(n, A019320(n)). - T. D. Noe, Apr 13 2010
a(n) = A019320(n)/(A019320(n) mod n) for n > 1. - Thomas Ordowski, Oct 24 2017

Extensions

More terms from Vladeta Jovovic, Apr 02 2004
Definition corrected by Jerry Metzger, Nov 04 2009

A064122 Number of divisors of 3^n - 1 that are relatively prime to 3^m - 1 for all 0 < m < n.

Original entry on oeis.org

2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 4, 2, 8, 4, 8, 4, 8, 2, 2, 8, 4, 2, 4, 4, 8, 2, 8, 4, 4, 4, 8, 2, 16, 8, 32, 4, 4, 4, 8, 4, 4, 4, 8, 8, 4, 2, 4, 4, 2, 2, 8, 4, 8, 4, 4, 2, 2, 2, 16, 8, 8, 4, 8, 16, 8, 4, 8, 4, 16, 4, 4, 2, 8, 8, 8, 4, 4, 4, 4, 8, 4, 4, 8, 4, 4, 8
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2001

Keywords

Links

Crossrefs

Cf. A063982.

Programs

  • Mathematica
    a = {1}; Do[ d = Divisors[ 3^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 3^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 100} ]
  • PARI
    { allocatemem(932245000); for (n=1, 167, d=divisors(3^n - 1); l=length(d); a=0; for (i=1, l, t=1; for (m=1, n - 1, p=3^m - 1; if (gcd(d[i], p)!=1, t=0; break)); if (t, a++)); write("b064122.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 08 2009

A064123 Number of divisors of 5^n - 1 that are relatively prime to 5^m - 1 for all 0 < m < n.

Original entry on oeis.org

3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 8, 4, 4, 8, 4, 2, 16, 4, 16, 2, 8, 4, 4, 4, 4, 4, 16, 4, 8, 8, 4, 4, 4, 8, 4, 4, 8, 4, 2, 2, 2, 4, 8, 4, 8, 8, 16, 2, 2, 4, 4, 4, 8, 8, 8, 8, 8, 4, 32, 16, 16, 4, 4, 8, 8, 8, 32, 4, 8, 4, 8, 4, 4, 16, 8, 4, 8, 16, 8, 2, 64, 2, 4, 2, 8, 8, 16, 4, 8, 8
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2001

Keywords

Links

Crossrefs

Cf. A063982.

Programs

  • Mathematica
    a = {1}; Do[ d = Divisors[ 5^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 5^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 58} ]
  • PARI
    { allocatemem(932245000); for (n=1, 119, d=divisors(5^n - 1); l=length(d); a=0; for (i=1, l, t=1; for (m=1, n - 1, p=5^m - 1; if (gcd(d[i], p)!=1, t=0; break)); if (t, a++)); write("b064123.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 08 2009

Extensions

More terms from Harry J. Smith, Sep 08 2009

A064124 Number of divisors of 7^n - 1 that are relatively prime to 7^m - 1 for all 0 < m < n.

Original entry on oeis.org

4, 1, 2, 3, 2, 2, 4, 2, 4, 4, 4, 4, 2, 4, 4, 4, 4, 2, 4, 4, 2, 4, 8, 8, 4, 4, 16, 2, 8, 2, 8, 4, 8, 2, 4, 2, 8, 4, 8, 4, 8, 4, 4, 8, 4, 2, 4, 2, 16, 2, 16, 4, 8, 2, 2, 16, 8, 2, 16, 4, 16, 8, 2, 8, 32, 4, 16, 8, 8, 16, 4, 8, 32, 4, 4, 2, 8, 8, 4, 8, 16, 16, 128, 2, 8
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2001

Keywords

Links

Crossrefs

Cf. A063982.

Programs

  • Mathematica
    a = {1}; Do[ d = Divisors[ 7^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 7^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 70} ]
  • PARI
    a(n) = sumdiv(7^n-1, d, vecsum(vector(n-1, k, gcd(d, 7^k-1) == 1)) == n-1); \\ Michel Marcus, Jun 23 2018

Extensions

a(71)-a(85) from Giovanni Resta, Jun 26 2018

A064126 Number of divisors of 10^n - 1 that are relatively prime to 10^m - 1 for all 0 < m < n.

Original entry on oeis.org

3, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 2, 8, 2, 4, 4, 4, 4, 2, 4, 8, 8, 2, 2, 8, 4, 4, 8, 32, 8, 8, 32, 4, 8, 8, 2, 8, 2, 2, 4, 16, 8, 16, 8, 4, 16, 4, 2, 4, 8, 16, 4, 16, 4, 16, 4, 8, 4, 4, 8, 128, 2, 32, 16, 4, 4, 8, 8, 8, 4, 4, 8, 8, 8, 8, 4, 16, 16, 64, 4, 32, 4
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2001

Keywords

Links

Crossrefs

Cf. A063982.

Programs

  • Mathematica
    a = {1}; Do[ d = Divisors[ 10^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 10^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 61} ]
  • PARI
    a(n) = sumdiv(10^n-1, d, vecsum(vector(n-1, k, gcd(d, 10^k-1) == 1)) == n-1); \\ Michel Marcus, Jun 23 2018

Extensions

a(62)-a(82) from Giovanni Resta, Jun 26 2018

A064127 Number of divisors of 11^n - 1 that are relatively prime to 11^m - 1 for all 0 < m < n.

Original entry on oeis.org

4, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 4, 4, 2, 2, 4, 2, 2, 2, 2, 8, 16, 8, 4, 8, 8, 2, 8, 4, 4, 8, 4, 8, 8, 4, 2, 16, 16, 64, 4, 128, 2, 4, 4, 2, 16, 8, 16, 8, 4, 16, 2, 16, 4, 16, 16, 4, 8, 8, 2, 8, 16, 32, 8, 4, 16, 16, 16, 8, 48, 32, 8, 2, 4, 16, 4, 4, 8, 8, 4
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2001

Keywords

Links

Crossrefs

Cf. A063982.

Programs

  • Mathematica
    a = {1}; Do[ d = Divisors[ 11^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 11^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 42} ]
  • PARI
    a(n) = sumdiv(11^n-1, d, vecsum(vector(n-1, k, gcd(d, 11^k-1) == 1)) == n-1); \\ Michel Marcus, Jun 23 2018

Extensions

a(43)-a(80) from Jon E. Schoenfield, Jun 23 2018

A064128 Number of divisors of 6^n - 1 that are relatively prime to 6^m - 1 for all 0 < m < n.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 16, 2, 4, 4, 2, 2, 16, 2, 4, 8, 4, 4, 2, 2, 4, 8, 4, 4, 8, 8, 32, 4, 4, 4, 4, 2, 16, 8, 8, 4, 8, 4, 8, 2, 8, 8, 8, 4, 4, 16, 8, 8, 8, 16, 8, 2, 16, 8, 16, 8, 4, 4, 4, 32, 2, 8, 16, 8, 8, 4, 4, 16, 8, 8, 4, 8, 128
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2001

Keywords

Links

Crossrefs

Cf. A063982.

Programs

  • Mathematica
    a = {1}; Do[ d = Divisors[ 6^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 6^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 66} ]
  • PARI
    a(n) = sumdiv(6^n-1, d, vecsum(vector(n-1, k, gcd(d, 6^k-1) == 1)) == n-1); \\ Michel Marcus, Jun 23 2018

Extensions

a(67)-a(83) from Jon E. Schoenfield, Jun 23 2018

A064129 Number of divisors of 12^n - 1 that are relatively prime to 12^m - 1 for all 0 < m < n.

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 4, 4, 4, 2, 4, 2, 4, 4, 8, 8, 4, 4, 2, 2, 2, 2, 8, 4, 4, 8, 4, 4, 16, 8, 16, 4, 8, 4, 32, 4, 4, 8, 8, 16, 8, 8, 16, 16, 16, 8, 4, 8, 4, 16, 4, 8, 64, 32, 8, 2, 16, 4, 8, 2, 8, 4, 2, 4, 16, 32, 16, 4, 4, 2, 8, 8, 4, 8, 8, 8, 32, 16, 8, 2, 8
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2001

Keywords

Links

Crossrefs

Cf. A063982.

Programs

  • Mathematica
    a = {1}; Do[ d = Divisors[ 12^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 12^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 46} ]
  • PARI
    a(n) = sumdiv(12^n-1, d, vecsum(vector(n-1, k, gcd(d, 12^k-1) == 1)) == n-1); \\ Michel Marcus, Jun 23 2018

Extensions

a(47)-a(81) from Jon E. Schoenfield, Jun 23 2018
Showing 1-8 of 8 results.

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

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