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.

A193295 Number of prime divisors (with multiplicity) of n^2 - 1.

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 3, 5, 3, 5, 2, 5, 3, 6, 3, 7, 2, 6, 3, 5, 3, 6, 3, 6, 5, 5, 4, 6, 2, 8, 3, 7, 4, 6, 3, 6, 3, 6, 3, 7, 2, 6, 4, 5, 4, 7, 3, 8, 4, 6, 3, 7, 3, 8, 4, 6, 3, 6, 2, 6, 4, 8, 5, 9, 3, 6, 3, 6, 3, 8, 2, 7, 4, 5, 5, 6, 3, 8, 5, 7, 5, 6, 3, 6, 4, 6
Offset: 2

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 22 2011

Keywords

Links

Crossrefs

Cf. A046051, A054992, A054990, A054991, A082863.

Programs

A240975 The number of distinct prime factors of n^3-1.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 3, 4, 2, 4, 3, 3, 2, 4, 3, 4, 3, 2, 3, 4, 4, 4, 2, 4, 3, 3, 3, 4, 3, 4, 4, 4, 3, 4, 2, 4, 3, 4, 2, 4, 4, 3, 4, 3, 3, 5, 2, 4, 4, 3, 3, 5, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 5, 2
Offset: 1

Views

Author

R. J. Mathar, Aug 05 2014

Keywords

Examples

			3^3-1 = 26 = 2*13, so a(3) = 2.
0 has no prime factors, so a(1) = 0.

Links

Crossrefs

Cf. A001221, A068601, A082863, A245909.

Programs

  • Maple
    A240975 := proc(n)
    A001221(n^3-1) ;
end proc:
  • Mathematica
    a[n_] := PrimeNu[n^3-1]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Sep 13 2024 *)
  • PARI
    a(n) = if(n<=1,0,omega(n^3-1)); \\ Joerg Arndt, Aug 06 2014
  • Python
    from sympy import primefactors
def A240975(n):
    return len(primefactors(n**3-1)) # Chai Wah Wu, Aug 06 2014

Formula

a(prime(n)) = A245909(n).
a(n) = A001221(A068601(n)) for n >= 2. - Michel Marcus, Aug 06 2014

A245908 The number of distinct prime factors of prime(n)^2-1.

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Aug 05 2014

Keywords

Links

Crossrefs

Cf. A008334, A008335, A082863.

Programs

  • Magma
    [#PrimeDivisors(NthPrime(n)^2 -1): n in [1..100]]; // Vincenzo Librandi, Apr 27 2019
  • Maple
    A245908 := proc(n)
    A082863(ithprime(n)) ;
end proc:
  • Mathematica
    Table[PrimeNu[Prime[n]^2 - 1], {n, 100}] (* Wesley Ivan Hurt, Aug 05 2014 *)
  • PARI
    vector(100, n, omega(prime(n)^2-1)) \\ Derek Orr, Aug 05 2014

Formula

a(n) = A082863(prime(n)).
a(n) = A008334(n) + A008335(n) - 1, if n>1.

A355177 Numbers k such that omega(k^2 - 1) = omega(k) where omega(k) = A001221(k).

Original entry on oeis.org

2, 3, 6, 10, 12, 15, 18, 24, 26, 28, 33, 48, 63, 66, 70, 72, 78, 80, 82, 84, 90, 105, 108, 110, 114, 120, 130, 132, 140, 156, 165, 170, 174, 182, 190, 192, 195, 222, 234, 238, 242, 252, 258, 264, 273, 276, 280, 290, 294, 306, 308, 310, 315, 318, 336, 342, 345, 350, 354, 357, 366, 372, 374, 378, 380, 385, 396, 399
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 22 2022

Keywords

Examples

			10 is in the sequence because A001221(10^2 - 1) = A001221(3*3*11) = 2 and A001221(10) = A001221(2*5) = 2.

Crossrefs

Cf. A001221, A082863, A190789.

Programs

  • Magma
    [n: n in [2..400] | #PrimeDivisors(n^2-1) eq #PrimeDivisors(n)];
  • Mathematica
    Select[Range[400], PrimeNu[#] == PrimeNu[#^2 - 1] &] (* Amiram Eldar, Jun 22 2022 *)
  • PARI
    is(n)=omega(n)==omega(n-1)+omega(n+1)-(n%2) \\ Charles R Greathouse IV, Jul 01 2022

Formula

A001221(a(n)) = A082863(a(n)).
Showing 1-4 of 4 results.

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