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.

A273284 A273282(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, 8, 19, 8, 9, 9, 23, 16, 25, 25, 27, 27, 29, 27, 31, 32, 25, 25, 25, 16, 37, 25, 25, 16, 41, 27, 43, 27, 27, 25, 47, 32, 49, 27, 49, 27, 53, 16, 49, 16, 49, 49, 59, 16, 61, 49, 27, 64, 49, 27, 67, 27, 49, 27, 71, 32, 73, 49
Offset: 2

Views

Author

Giuseppe Coppoletta, May 20 2016

Keywords

Comments

a(n) <= A079867(n) for any n>=2.
a(n) = n iff n is the power of a prime (A246655).

Examples

			a(33) = 25 because Omega(33)=2 and 5^2 < 33 < 7^2.
If n= 3^3 * 31^2 * 67 then a(n)= 7^6 and A273285(n)=11^6 because Omega(n)=6 and 7^6 < n < 11^6.

Links

Crossrefs

Cf. A273282, A273285, A273290, A079867, A246655, A001222.

Programs

  • Mathematica
    Table[NextPrime[(Floor[n^(1/PrimeOmega[n])] + 1) , -1]^PrimeOmega[n], {n,2,50}] (* G. C. Greubel, May 26 2016 *)
  • PARI
    a(n) = my(bn=bigomega(n)); precprime(sqrtnint(n, bn))^bn; \\ Michel Marcus, May 24 2016
  • Sage
    s=sloane.A001222; [previous_prime(floor(n^(1/s(n)))+1)^s(n) for n in (2..74)]

Formula

a(n) = A273282(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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