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.

A273290 A273288(n)^Omega(n), where Omega = A001222.

Original entry on oeis.org

2, 3, 4, 5, 4, 7, 8, 9, 9, 11, 8, 13, 9, 9, 16, 17, 27, 19, 8, 25, 25, 23, 16, 25, 49, 27, 8, 29, 27, 31, 32, 49, 49, 25, 16, 37, 49, 49, 16, 41, 27, 43, 8, 27, 121, 47, 32, 49, 125, 49, 8, 53, 81, 49, 16, 121, 169, 59, 16, 61, 169, 27, 64, 49, 27, 67, 8, 169, 125
Offset: 2

Views

Author

Giuseppe Coppoletta, May 25 2016

Keywords

Comments

a(n) is by definition the power of a prime. It coincides with n iff n is the power of a prime (A246655).
a(n) <= A273291(n)

Examples

			a(70) = A273291(70) = 5^3 because the median  of its prime factors [2, 5, 7] is the central value 5 (prime) and Omega(70)=3.
a(308) = 3^4 because Omega(308)=4 and the median of [2, 2, 7, 11] is (2+7)/2 = 4.5, whose previous prime is 3.

Links

Crossrefs

Cf. A273284, A273288, A273291, A079867, A246655, A001222.

Programs

  • Mathematica
    Table[Prime[PrimePi@ Median@ #]^Length@ # &@ Flatten@ Apply[Table[#1 {#2}] &, FactorInteger@ n, 1], {n, 2, 75}] (* Michael De Vlieger, May 27 2016 *)
  • Sage
    def pfwr(n): return flatten([([p] * m) for (p, m) in factor(n)]) # (list of prime factors of n with repetition)
[previous_prime(floor(median(pfwr(n)))+1)^sloane.A001222(n) for n in (2..70)]

Formula

a(n) = A273288(n)^A001222(n).

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)).

A273289 Least prime not less than the median of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

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

Views

Author

Giuseppe Coppoletta, May 25 2016

Keywords

Comments

A273288(n)<= a(n)<= A006530<= n and a(n) = n iff n is prime.

Examples

			a(76) = 2 because the median of its prime factors [2, 2, 19] is the central value 2 (and it is prime).
a(308) = 5 because the median of [2, 2, 7, 11] is commonly defined as the mean of the central values (2+7)/2 = 4.5 and the next prime is 5.

Links

Crossrefs

Cf. A273283, A273285, A273288, A273291, A079870, A006530.

Programs

  • Mathematica
    Table[If[PrimeQ@ #, #, NextPrime@ #] &@ 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])]; [next_prime(ceil(median(r(n)))-1) for n in (2..100)]
Showing 1-3 of 3 results.

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