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 1-4 of 4 results.

A144755 Primes which divide none of overpseudoprimes to base 2 (A141232).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, 2731, 5419, 8191, 43691, 61681, 65537, 87211, 131071, 174763, 262657, 524287, 599479, 2796203, 15790321, 18837001, 22366891, 715827883, 2147483647, 4278255361
Offset: 1

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Author

Vladimir Shevelev, Sep 20 2008

Keywords

Comments

Odd prime p is in the sequence iff A064078(A002326((p-1)/2))=p. For example, for p=127 we have A002326((127-1)/2)=7 and A064078(7)=127. Thus p=127 is in the sequence.
Primes p such that the binary expansion of 1/p has a unique period length; that is, no other prime has the same period. Sequence A161509 sorted. - T. D. Noe, Apr 13 2010
Since A161509 has terms of varying magnitude, sorting any finite initial segment of A161509 cannot provide a guarantee that there are no other terms missed in between. Any prime p not (yet) appearing in A161509 should be tested via A064078(A002326((p-1)/2))=p to conclude whether it belongs to the current sequence. - Max Alekseyev, Feb 10 2024

Examples

			Overpseudoprimes to base 2 are odd, then a(1)=2.

Crossrefs

Cf. A141232, A002326, A064078, A112927.
Cf. A040017 (unique-period primes in base 10). - T. D. Noe, Apr 13 2010

Programs

  • Mathematica
    b=2; t={}; Do[c=Cyclotomic[n,b]; q=c/GCD[n,c]; If[PrimePowerQ[q], p=FactorInteger[q][[1,1]]; If[p<10^12, AppendTo[t,p]; Print[{n,p}]]], {n,1000}]; t=Sort[t] (* T. D. Noe, Apr 13 2010 *)
  • PARI
    { is_a144755(p) = my(q,m,g); q=znorder(Mod(2,p)); m=2^q-1; fordiv(q,d, if(d1,m\=g))); m==p; } \\ Max Alekseyev, Feb 10 2024

Extensions

Extended by T. D. Noe, Apr 13 2010
b-file deleted by Max Alekseyev, Feb 10 2024.

A161508 Numbers k such that 2^k-1 has only one primitive prime factor.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208
Offset: 1

Views

Author

T. D. Noe, Jun 17 2009

Keywords

Comments

Also, numbers k such that A086251(k) = 1.
Also, numbers k such that A064078(k) is a prime power.
The corresponding primitive primes are listed in A161509.
The binary expansion of 1/p has period k and this is the only prime with such a period. The binary analog of A007498.
This sequence has many terms in common with A072226. A072226 has the additional term 6; but it does not have terms 18, 20, 21, 54, 147, 342, 602, and 889 (less than 10000).
All known terms that are not in A072226 belong to A333973.

Links

Crossrefs

Cf. A007498, A064078, A072226, A086251, A144755, A161509, A247071, A333973.

Programs

  • Mathematica
    Select[Range[1000], PrimePowerQ[Cyclotomic[ #,2]/GCD[Cyclotomic[ #,2],# ]]&]
  • PARI
    is_A161508(n) = my(t=polcyclo(n,2)); isprimepower(t/gcd(t,n)); \\ Charles R Greathouse IV, Nov 17 2014

A247071 Numbers n such that 2^n-1 has only one primitive prime factor, sorted according to the magnitude of the corresponding prime.

Original entry on oeis.org

2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, 26, 42, 13, 34, 40, 32, 54, 17, 38, 27, 19, 33, 46, 56, 90, 78, 62, 31, 80
Offset: 1

Views

Author

Eric Chen, Nov 16 2014

Keywords

Comments

Periods associated with A144755 in base 2. The binary analog of A051627.

Examples

			2^12 - 1 = 4095 = 3 * 3 * 5 * 7 * 13, but none of 3, 5, 7 is a primitive prime factor, so the only primitive prime factor of 2^12 - 1 is 13.

Crossrefs

Cf. A161508, A161509, A144755, A007498, A007615, A051627, A040017.

Programs

  • Mathematica
    nmax = 65536; primesPeriods = Reap[Do[p = Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]]

Formula

a(n) = A002326((A144755(n+1)-1)/2). - Max Alekseyev, Feb 11 2024

Extensions

Sequence trimmed to the established terms of A144755 by Max Alekseyev, Feb 11 2024

A333973 Numbers k such that A019320(k) is greater than A064078(k) and the latter is a prime or a prime power.

Original entry on oeis.org

18, 20, 21, 54, 147, 342, 602, 889, 258121
Offset: 1

Views

Author

Jeppe Stig Nielsen, Sep 22 2020

Keywords

Comments

The unique prime factor of A064078(k) is then a unique prime to base 2 (see A161509), but not a cyclotomic number.
Subsequence of A161508. In fact, subsequence of the set difference A161508 \ A072226.
In all known examples, A064078(k) is a prime. If A064078(k) was a prime power p^j with j>1, then p would be both a Wieferich prime (A001220) and a unique prime to base 2.
Subsequence of A093106 (the characterization of A093106 can be useful when searching for more terms).
Should this sequence be infinite?

Links

Crossrefs

Cf. A019320, A064078, A093106, A072226, A144755, A161508, A161509, A247071.

Programs

  • PARI
    for(n=1,+oo,c=polcyclo(n,2); c % n < 2 && next(); c/=(c%n); ispseudoprime(if(ispower(c,,&b),b,c))&&print1(n, ", "))
Showing 1-4 of 4 results.

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