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 1-6 of 6 results.

A158584 Erroneous version of A079866.

Original entry on oeis.org

2, 3, 1, 5, 2, 7, 1, 3, 3, 11, 2, 13, 3, 3, 1, 17, 2, 19, 2, 4, 4, 23, 2, 4, 5, 3, 3, 29, 3, 31, 1, 5, 5, 5, 2, 37, 6, 6, 2, 41, 3, 43, 3, 3, 6, 47, 2, 7, 3, 7, 3, 53, 2, 7, 2, 7, 7, 59, 2, 61, 7, 3, 1, 8, 4, 67, 4, 8, 4, 71, 2, 73, 8, 4, 4, 8, 4, 79, 2, 3, 9, 83, 3, 9, 9, 9, 3, 89, 3, 9, 4, 9, 9, 9, 2, 97
Offset: 2

Views

Author

Cino Hilliard, Mar 21 2009

Keywords

Comments

We do not begin with the unit 1 because it has no prime factors. Conjecture: The sequence contains the set of prime numbers more than once.

Examples

			12=2*2*3 has 3 factors; 12^(1/3) = 2.289428... so 2 is in the 11th position in the sequence.

Programs

  • PARI
    g(n) = for(x=2,n,print1(floor(x^(1/bigomega(x)))","))

Formula

The geometric mean is the n-th root of the product of n numbers.
Gm = (a(1)*a(2)*...*a(n))^(1/n).

A079867 a(1)=1 and for n>1: floor(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 9, 11, 8, 13, 9, 9, 16, 17, 8, 19, 8, 16, 16, 23, 16, 25, 25, 27, 27, 29, 27, 31, 32, 25, 25, 25, 16, 37, 36, 36, 16, 41, 27, 43, 27, 27, 36, 47, 32, 49, 27, 49, 27, 53, 16, 49, 16, 49, 49, 59, 16, 61, 49, 27, 64, 64, 64, 67, 64, 64, 64, 71, 32, 73, 64
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 13 2003

Keywords

Comments

a(n)<=A079869(n); A020639(n)<=a(n)<=A006530(n);
a(m)=m=A079869(m)=A079871(m) iff m is a prime power (A000961).

Crossrefs

a(n)=A079866(n)^A001222(n), cf. A068794, A068795.

Programs

  • Mathematica
    Join[{1},Table[Floor[n^(1/PrimeOmega[n])]^PrimeOmega[n],{n,2,80}]] (* Harvey P. Dale, May 19 2018 *)

A273282 Largest prime not exceeding the geometric mean of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 2, 3, 3, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 5, 5, 2, 41, 3, 43, 3, 3, 5, 47, 2, 7, 3, 7, 3, 53, 2, 7, 2, 7, 7, 59, 2, 61, 7, 3, 2, 7, 3, 67, 3, 7, 3, 71, 2, 73, 7, 3, 3, 7, 3, 79, 2, 3, 7
Offset: 2

Views

Author

Giuseppe Coppoletta, May 19 2016

Keywords

Comments

a(n) = n iff n is prime.
a(n) <= A079866(n) with equality iff A079866(n) is prime.

Examples

			a(46) = 5 because 5 is the greatest prime not bigger than sqrt(2*23).
For n = 5^3 * 11 * 89, a(n)=7 and A273283(n)=11 because A001222(n)=5 and 7 < n^(1/5) < 11.

Links

Crossrefs

Cf. A273283, A273284, A273288, A079866, A001222.

Programs

  • Mathematica
    a[n_] := NextPrime[ Floor[n^ (1/PrimeOmega[n])] + 1, -1]; a /@ Range[2, 100] (* Giovanni Resta, May 25 2016 *)
  • PARI
    a(n) = precprime(sqrtnint(n, bigomega(n))); \\ Michel Marcus, May 24 2016
  • Sage
    [previous_prime(floor(n^(1/sloane.A001222(n)))+1) for n in (2..100)]

Formula

For n>=2, a(n) = A007917(A079866(n)).

A273288 Largest prime not exceeding the median of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 3, 19, 2, 5, 5, 23, 2, 5, 7, 3, 2, 29, 3, 31, 2, 7, 7, 5, 2, 37, 7, 7, 2, 41, 3, 43, 2, 3, 11, 47, 2, 7, 5, 7, 2, 53, 3, 7, 2, 11, 13, 59, 2, 61, 13, 3, 2, 7, 3, 67, 2, 13, 5, 71, 2, 73, 19, 5, 2, 7, 3, 79, 2, 3, 19
Offset: 2

Views

Author

Giuseppe Coppoletta, May 25 2016

Keywords

Comments

A020639(n)<= a(n)<= A273289(n).
a(n) = n iff n is prime.

Examples

			a(66) = 3 because the median of [2, 3, 11] is the central value 3 (and it is prime).
a(308) = 3 because the median of [2, 2, 7, 11] is (2+7)/2 = 4.5 and the previous prime is 3.

Links

Crossrefs

Cf. A273282, A273284, A273289, A273290, A079866, A020639.

Programs

  • Mathematica
    Table[Prime@ PrimePi@ Median@ Flatten@ Apply[Table[#1, {#2}] &, FactorInteger@ n, 1], {n, 2, 82}] (* Michael De Vlieger, May 27 2016 *)
  • Sage
    r = lambda n: [f[0] for f in factor(n) for _ in range(f[1])]; [previous_prime(floor(median(r(n)))+1) for n in (2..100)]

A079868 a(1)=1 and for n>1: round(n^(1/Omega(n))), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 3, 19, 3, 5, 5, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 6, 6, 6, 2, 37, 6, 6, 3, 41, 3, 43, 4, 4, 7, 47, 2, 7, 4, 7, 4, 53, 3, 7, 3, 8, 8, 59, 3, 61, 8, 4, 2, 8, 4, 67, 4, 8, 4, 71, 2, 73, 9, 4, 4, 9, 4, 79, 2, 3, 9, 83, 3, 9, 9, 9, 3, 89, 3, 10, 5, 10, 10
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 13 2003

Keywords

Comments

A079866(n)<=a(n)<=A079870(n); A020639(n)<=a(n)<=A006530(n);
a(m)=A079866(m)=A079870(m) iff m is a prime power (A000961).

Crossrefs

Cf. A079869(n)=a(n)^A001222(n), A079881.

Programs

  • Mathematica
    Join[{1},Table[Floor[n^(1/PrimeOmega[n])+1/2],{n,2,100}]] (* Harvey P. Dale, Aug 11 2012 *)

A276632 Integer part of the geometric mean of the prime factors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 3, 4, 4, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 6, 6, 3, 41, 3, 43, 4, 3, 6, 47, 2, 7, 3, 7, 5, 53, 2, 7, 3, 7, 7, 59, 3, 61, 7, 4, 2, 8, 4, 67, 5, 8, 4, 71, 2, 73, 8, 3, 6, 8, 4, 79, 3
Offset: 1

Views

Author

R. J. Mathar, Sep 08 2016

Keywords

Examples

			For n=20, the two distinct prime factors are 2 and 5, the arithmetic mean is sqrt(2*5), and the integer part is a(20)=3.

Links

Crossrefs

Cf. A079866 (primes with multiplicity)

Programs

  • Maple
    A276632 := proc(n)
    floor(root[A001221(n)](A007947(n))) ;
end proc:
seq(A276632(n),n=1..80) ;
  • Mathematica
    rad[n_] := Times @@ (First@# & /@ FactorInteger@n); Table[Floor[(rad[n])^(1/PrimeNu[n])], {n, 1, 50}] (* G. C. Greubel, May 19 2017 *)

Formula

a(n) = floor( A007947(n)^(1/A001221(n)) ).
Showing 1-6 of 6 results.

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