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.

A273291 A273289(n)^Omega(n), where Omega = A001222.

Original entry on oeis.org

2, 3, 4, 5, 9, 7, 8, 9, 25, 11, 8, 13, 25, 25, 16, 17, 27, 19, 8, 25, 49, 23, 16, 25, 121, 27, 8, 29, 27, 31, 32, 49, 121, 49, 81, 37, 121, 121, 16, 41, 27, 43, 8, 27, 169, 47, 32, 49, 125, 121, 8, 53, 81, 121, 16, 121, 289, 59, 81, 61, 289, 27, 64, 121, 27, 67, 8, 169, 125
Offset: 2

Views

Author

Giuseppe Coppoletta, May 25 2016

Keywords

Comments

a(n)>= A273290(n).
a(n) is by definition the power of a prime. It coincides with n iff n is also a power of prime (A246655).

Examples

			a(308) = 5^4 because Omega(308)=4 and the median of [2, 2, 7, 11] is (2+7)/2 = 4.5, whose next prime is 5. See A273290 for other examples.

Links

Crossrefs

Cf. A273285, A273289, A273290, A079871, A001222, A246655.

Programs

  • Mathematica
    Table[If[PrimeQ@ #, #, NextPrime@ #] &[Median@ #]^Length@ # &@ Flatten@ Apply[Table[#1, {#2}] &, FactorInteger@ n, 1], {n, 2, 70}] (* Michael De Vlieger, May 27 2016 *)
  • Sage
    def r(n): return [f[0] for f in factor(n) for _ in range(f[1])]
[next_prime(ceil(median(r(n)))-1)^sloane.A001222(n) for n in (2..70)]

Formula

a(n) = A273289(n)^A001222(n).

A273283 Least prime not less than the geometric mean of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

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

Views

Author

Giuseppe Coppoletta, May 19 2016

Keywords

Comments

A079870(n) <= a(n) <= A006530(n) <= n and a(n) = n iff n is prime, while a(n)= A079870(n) iff A079870(n) is prime.

Examples

			a(46)=7 because 7 is the least prime not less than sqrt(2*23).
a(84)=5 and A273282(84)=3 because A001222(84)=4 and 3 < 84^(1/4) < 5.

Links

Crossrefs

Cf. A273282, A273285, A273289, A079870, A079871, A006530, A001222.

Programs

  • Mathematica
    Table[NextPrime[(Ceiling[n^(1/PrimeOmega[n])] - 1)], {n,2,50} ] (* G. C. Greubel, May 26 2016 *)
  • Sage
    [next_prime(ceil(n^(1/sloane.A001222(n)))-1) for n in (2..82)]

Formula

For n >= 2, a(n) = A007918(A079870(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)]
Showing 1-3 of 3 results.

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