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 1-3 of 3 results.

A191592 Numbers in A191311 but not in A129521.

Original entry on oeis.org

4, 11305, 13981, 23001, 30889, 39865, 68101, 88561, 91001, 93961, 107185, 137149, 152551, 157641, 176149, 204001, 228241, 251251, 276013, 401401, 464185, 493697, 493885, 534061, 563473, 574561, 622909, 631351, 683761, 786961, 915981, 950797, 1106785, 1141141
Offset: 1

Views

Author

Jason Holt, Jun 04 2011

Keywords

Links

Crossrefs

Cf. A191311, A129521.

Extensions

More terms from David W. Wilson, Aug 16 2011

A129521 Numbers of the form p*q, p and q prime with q=2*p-1.

Original entry on oeis.org

6, 15, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 19 2007

Keywords

Comments

All terms are Fermat 4-pseudoprimes, i.e., satisfy 4^n == 4 (mod n). See A020136 and A122781.

Links

Crossrefs

Cf. A005382, A005383.
Subsequence of A006881, A129510, and A122781.
Intersection of A000384 and A001358, "hexagonal semiprimes". - Wesley Ivan Hurt, Jul 04 2013
Cf. A141768, A191311, A191592, A245365, A259676, A259677.

Programs

  • Haskell
    a129521 n = p * (2 * p - 1) where p = a005382 n
-- Reinhard Zumkeller, Nov 10 2013
  • Magma
    [2*n^2-n: n in [0..1000]|IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 27 2010
  • Mathematica
    p = Select[Prime[Range[155]], PrimeQ[2# - 1] &]; p (2p - 1) (* Robert G. Wilson v, Sep 11 2011 *)
  • PARI
    forprime(p=2,10000,q=2*p-1;if(isprime(q),print1(p*q,", ")))

Formula

a(n) = A005382(n)*A005383(n).

A247074 a(n) = phi(n)/(Product_{primes p dividing n } gcd(p - 1, n - 1)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 1, 6, 2, 8, 1, 6, 1, 8, 3, 10, 1, 8, 5, 12, 9, 4, 1, 8, 1, 16, 5, 16, 6, 12, 1, 18, 6, 16, 1, 12, 1, 20, 3, 22, 1, 16, 7, 20, 8, 8, 1, 18, 10, 24, 9, 28, 1, 16, 1, 30, 9, 32, 3, 4, 1, 32, 11, 8, 1, 24, 1, 36, 10, 12, 15, 24, 1, 32, 27, 40, 1, 24, 4, 42, 14, 40, 1, 24, 2, 44, 15, 46
Offset: 1

Views

Author

Eric Chen, Nov 16 2014

Keywords

Comments

a(n) = A000010(n)/A063994(n). - Eric Chen, Nov 29 2014
Does every natural number appear in this sequence? If so, do they appear infinitely many times? - Eric Chen, Nov 26 2014
A063994(n) must be a factor of EulerPhi(n). - Eric Chen, Nov 26 2014
Number n is (Fermat) pseudoprimes (or prime) to one in a(n) of its coprime bases. That is, b^(n-1) = 1 (mod n) for one in a(n) of numbers b coprime to n. - Eric Chen, Nov 26 2014
a(n) = 1 if and only if n is 1, prime (A000040), or Carmichael number (A002997). - Eric Chen, Nov 26 2014
a(A191311(n)) = 2. - Eric Chen, Nov 26 2014
a(p^n) = p^(n-1), where p is a prime. - Eric Chen, Nov 26 2014
a(A209211(n)) = EulerPhi(A209211(n)), because A063994(A209211(n)) = 1. - Eric Chen, Nov 26 2014

Examples

			EulerPhi(15) = 8, and that 15 is a Fermat pseudoprime in base 1, 4, 11, and 14, the total is 4 bases, so a(15) = 8/4 = 2.

Links

Crossrefs

Cf. A063994, A182816, A002997, A191311, A064234, A000010, A063994, A003557, A209211.

Programs

  • Mathematica
    a063994[n_] := Times @@ GCD[n - 1, First /@ FactorInteger@ n - 1]; a063994[1] = 1; a247074[n_] := EulerPhi[n]/a063994[n]; Array[a247074, 150]
  • PARI
    a(n)=my(f=factor(n));eulerphi(f)/prod(i=1,#f~,gcd(f[i,1]-1,n-1)) \\ Charles R Greathouse IV, Nov 17 2014

Formula

A003557(n) <= a(n) <= n, and a(n) is a multiple of A003557(n). - Charles R Greathouse IV, Nov 17 2014
Showing 1-3 of 3 results.

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