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.

A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

Original entry on oeis.org

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

Views

Author

Andrew S. Plewe, Feb 20 2004

Keywords

Comments

Omega(n) of the odd integers follows a pattern similar to A001222, with 4 maxima instead of 2 - i.e. between 2^n and (2^(n+1) - 1) there are two numbers with exactly n factors (2^n and 2^(n-1) * 3) while the odd integers have 4 maxima (3^n, 3^(n-1) * 5, 3^(n-1) * 7, 5^2*3^(n-2)) between 3^n and 3^(n+1) - 1.

Examples

			Omega(1) = 0, Omega(9) = 2 (3 * 3 = 9), Omega (243) = 5 (3 * 3 * 3 * 3 * 3 = 243), Omega(51) = 2 (3 * 17 = 51).
For n = 92, A001222(2*92 - 1) = A001222(183) = 2 as 183 = 3*61, thus a(92) = 2. - _Antti Karttunen_, May 31 2017

Links

Crossrefs

One more than A285716 (after the initial term).
Cf. A001222, A000120, A092523, A099774, A099990, A244153, A278223, A286582.
Cf. A006254 (positions of ones).

Programs

  • Mathematica
    a[n_] := PrimeOmega[2*n - 1]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)
  • PARI
    a(n) = bigomega(2*n-1) \\ Michel Marcus, Jul 26 2013, edited to reflect the changed starting offset by Antti Karttunen, May 31 2017

Formula

a(n) = Omega(2n-1). [Odd bisection of A001222.]
From Antti Karttunen, May 31 2017: (Start)
For n >= 1, a(n) = A000120(A244153(n)).
For n >= 2, a(n) = 1+A285716(n).
(End)

Extensions

Starting offset changed to 1 and the definition modified respectively. Also values of the initial term and of term a(92) (= 2, previously a(91) = 1) corrected by Antti Karttunen, May 31 2017

A099812 Number of distinct primes dividing 2n (i.e., omega(2n)).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Nov 19 2004

Keywords

Comments

Bisection of A001221.

Examples

			a(6) = 2 because 12 = 2*2*3 has 2 distinct prime divisors.
a(15) = 3 because 30 = 2*3*5 has 3 distinct prime divisors.

Links

Crossrefs

Cf. A001221, A077761, A092523.

Programs

  • Magma
    [#PrimeDivisors(2*n): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017
  • Maple
    with(numtheory): omega:=proc(n) local div,A,j: div:=divisors(n): A:={}: for j from 1 to tau(n) do if isprime(div[j])=true then A:=A union {div[j]} else A:=A fi od: nops(A) end: seq(omega(2*n),n=1..130); # Emeric Deutsch, Mar 10 2005
  • Mathematica
    Table[PrimeNu[2*n], {n,1,50}] (* G. C. Greubel, May 21 2017 *)
  • PARI
    for(n=1,50, print1(omega(2*n), ", ")) \\ G. C. Greubel, May 21 2017

Formula

From Amiram Eldar, Sep 21 2024: (Start)
a(n) = A001221(2*n).
a(n) = omega(n) + 1 if n is odd, and a(n) = omega(n) if n is even.
Sum_{k=1..n} a(k) = n * (log(log(n)) + B + 1/2) + O(n/log(n)), where B is Mertens's constant (A077761). (End)

Extensions

More terms from Emeric Deutsch, Mar 10 2005

A120891 Number of primitive Pythagorean triangles with odd leg 2n-1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 4, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 4, 1, 1, 2, 2
Offset: 1

Views

Author

Lekraj Beedassy, Jul 12 2006

Keywords

Comments

Bisection of A024361.
Bisection of even-numbered terms of A024361 results in alternating zero terms; removing zeros gives A068068. - Ray Chandler, Feb 04 2020

Links

Crossrefs

Cf. A024361, A068068, A092523, A120890.

Formula

a(n)=2^(k-1), where k=A092523(n) for n > 1.

Extensions

a(1)=0 inserted by Ray Chandler, Feb 04 2020

A192688 Sum of omega-values for two consecutive indices where they equal the omega-value at the sum of the two indices.

Original entry on oeis.org

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

Views

Author

Juri-Stepan Gerasimov, Jul 07 2011

Keywords

Comments

The sequence contains sums A001221(n) + A001221(n+1) whenever they are equal to A001221(2n+1).
Generated by indices n = 1, 7, 16, 31, 52, 82, 97, 136, 157, 172, 178, 192, 199, 232, 241, 256, 262, 277, 292, ...

Crossrefs

Cf. A001221, A059957, A092523.

Programs

  • Maple
    for n from 1 to 2000 do
        if A001221(n)+A001221(n+1) = A001221(2*n+1) then
                printf("%d,",A001221(2*n+1)) ;
        end if;
end do: # R. J. Mathar, Oct 12 2011
  • Mathematica
    f[n_] := Block[{a = PrimeNu[n] + PrimeNu[n + 1]}, If[a == PrimeNu[2n + 1], Return@ a]]; k = 1; lst = {}; While[k < 2600, If[f@k > 0, AppendTo[lst, f@ k]]; k++]; lst (* Robert G. Wilson v, Aug 29 2011 *)
Showing 1-4 of 4 results.

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

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