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.

A079869 a(1)=1 and for n>1: round(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, 16, 16, 16, 17, 27, 19, 27, 25, 25, 23, 16, 25, 25, 27, 27, 29, 27, 31, 32, 36, 36, 36, 16, 37, 36, 36, 81, 41, 27, 43, 64, 64, 49, 47, 32, 49, 64, 49, 64, 53, 81, 49, 81, 64, 64, 59, 81, 61, 64, 64, 64, 64, 64, 67, 64, 64, 64, 71, 32, 73
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 13 2003

Keywords

Comments

A079867(n)<=a(n)<=A079869(n); A020639(n)<=a(n)<=A006530(n);
a(m)=m=A079867(m)=A079871(m) iff m is a prime power (A000961).

Crossrefs

a(n)=A079868(n)^A001222(n). Cf. A079872, A068794, A068795.

Programs

  • Mathematica
    ron[n_]:=Module[{c=PrimeOmega[n]},Round[n^(1/c)]^c]; Join[{1},Array[ ron,80,2]] (* Harvey P. Dale, Jun 17 2020 *)

A079870 a(1)=1 and for n>1: ceiling(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, 3, 7, 2, 3, 4, 11, 3, 13, 4, 4, 2, 17, 3, 19, 3, 5, 5, 23, 3, 5, 6, 3, 4, 29, 4, 31, 2, 6, 6, 6, 3, 37, 7, 7, 3, 41, 4, 43, 4, 4, 7, 47, 3, 7, 4, 8, 4, 53, 3, 8, 3, 8, 8, 59, 3, 61, 8, 4, 2, 9, 5, 67, 5, 9, 5, 71, 3, 73, 9, 5, 5, 9, 5, 79, 3, 3, 10, 83, 4, 10, 10, 10, 4, 89, 4, 10, 5, 10
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 13 2003

Keywords

Links

Crossrefs

Cf. A001222, A006530, A020639, A068794, A068795, A079869, A079871.

Programs

  • Mathematica
    A079870[n_] := If [n == 1, 1, Ceiling[n^(1/PrimeOmega[n])]];
Array[A079870, 100] (* Paolo Xausa, Oct 28 2024 *)
  • PARI
    a(n) = if (n==1, 1, ceil(n^(1/bigomega(n)))); \\ Michel Marcus, May 31 2016

Formula

A079871(n) = a(n)^A001222(n).
a(n) >= A079868(n); A020639(n) <= a(n) <= A006530(n);
a(m) = A079868(m) iff m is a prime power (A000961).

A273285 A273283(n)^Omega(n), where Omega = A001222.

Original entry on oeis.org

2, 3, 4, 5, 9, 7, 8, 9, 25, 11, 27, 13, 25, 25, 16, 17, 27, 19, 27, 25, 25, 23, 81, 25, 49, 27, 125, 29, 125, 31, 32, 49, 49, 49, 81, 37, 49, 49, 81, 41, 125, 43, 125, 125, 49, 47, 243, 49, 125, 121, 125, 53, 81, 121, 81, 121, 121, 59, 81, 61, 121, 125, 64, 121, 125, 67, 125, 121, 125, 71
Offset: 2

Views

Author

Giuseppe Coppoletta, May 20 2016

Keywords

Comments

a(n) >= A079871(n) for any n>=2.
a(n) = n iff n is the power of a prime (A246655).

Examples

			a(33) = 49 because Omega(33)=2 and 5^2 < 33 < 7^2.
a(2403) = 11^4 (and A273284(2403)=7^4) because A001222(3^3*89)=4 and 7^4 < n < 11^4.

Links

Crossrefs

Cf. A273283, A273284, A273291, A079871, A001222, A246655.

Programs

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

Formula

a(n) = A273283(n)^A001222(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 *)

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)).

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).
Showing 1-6 of 6 results.

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