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 1-8 of 8 results.

A254732 a(n) is the least k > n such that n divides k^2.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 14, 12, 12, 20, 22, 18, 26, 28, 30, 20, 34, 24, 38, 30, 42, 44, 46, 36, 30, 52, 36, 42, 58, 60, 62, 40, 66, 68, 70, 42, 74, 76, 78, 60, 82, 84, 86, 66, 60, 92, 94, 60, 56, 60, 102, 78, 106, 72, 110, 84, 114, 116, 118, 90, 122, 124, 84, 72
Offset: 1

Views

Author

Peter Kagey, Feb 06 2015

Keywords

Comments

A073353(n) <= a(n) <= 2*n. Any prime that divides n must also divide a(n), and because n divides (2*n)^2.
Are all terms even? -Harvey P. Dale, Aug 07 2025

Examples

			a(12) = 18 because 12 divides 18^2, but 12 does not divide 13^2, 14^2, 15^2, 16^2, or 17^2.

Links

Crossrefs

Cf. A254733 (similar, with k^3), A254734 (similar, with k^4), A073353 (similar, with limit m->infinity of k^m).
Cf. A253905.

Programs

  • Haskell
    a254732 n = head [k | k <- [n + 1 ..], mod (k ^ 2) n == 0]
-- Reinhard Zumkeller, Feb 07 2015
  • Mathematica
    lk[n_]:=Module[{k=n+1},While[!Divisible[k^2,n],k++];k]; Array[lk,70] (* Harvey P. Dale, Nov 05 2017 *)
Table[Module[{k=n+1},While[PowerMod[k,2,n]!=0,k++];k],{n,70}] (* Harvey P. Dale, Aug 07 2025 *)
  • PARI
    a(n)=for(k=n+1,2*n,if(k^2%n==0,return(k)))
vector(100,n,a(n)) \\ Derek Orr, Feb 06 2015
  • PARI
    a(n)=my(t=factorback(factor(n)[,1])); forstep(k=n+t,2*n,t,if(k^2%n==0, return(k))) \\ Charles R Greathouse IV, Feb 07 2015
  • Python
    def A254732(n):
    k = n + 1
    while pow(k,2,n):
        k += 1
    return k # Chai Wah Wu, Feb 15 2015
  • Ruby
    def a(n)
  (n+1..2*n).find { |k| k**2 % n == 0 }
end

Formula

a(n) = sqrt(n*A072905(n)).
a(n) = A019554(n)*(A000188(n)+1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + zeta(3)/zeta(2) = 1 + A253905 = 1.73076296940143849872... . - Amiram Eldar, Feb 17 2024

A254733 a(n) is the least k > n such that n divides k^3.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 14, 10, 12, 20, 22, 18, 26, 28, 30, 20, 34, 24, 38, 30, 42, 44, 46, 30, 30, 52, 30, 42, 58, 60, 62, 36, 66, 68, 70, 42, 74, 76, 78, 50, 82, 84, 86, 66, 60, 92, 94, 60, 56, 60, 102, 78, 106, 60, 110, 70, 114, 116, 118, 90, 122, 124, 84
Offset: 1

Views

Author

Peter Kagey, Feb 06 2015

Keywords

Comments

A073353(n) <= a(n) <= 2*n. Any prime that divides n must also divide a(n), and because n divides (2*n)^3.

Examples

			a(8) = 10 because 8 divides 10^3, but 8 does not divide 9^3.

Links

Crossrefs

Cf. A073353 (similar, with k^n).
Cf. A254732 (similar, with k^2), A254734 (similar, with k^4).
Cf. A019555 (similar without the restriction that a(n) > n).

Programs

  • Mathematica
    lkn[n_]:=Module[{k=n+1},While[PowerMod[k,3,n]!=0,k++];k]; Array[lkn,70] (* Harvey P. Dale, Nov 23 2024 *)
  • PARI
    a(n)=for(k=n+1,2*n,if(k^3%n==0,return(k)))
vector(100,n,a(n)) \\ Derek Orr, Feb 07 2015
  • Ruby
    def a(n)
  (n+1..2*n).find { |k| k**3 % n == 0 }
end

Formula

a(n) = n + A019555(n).

A254734 a(n) is the least k > n such that n divides k^4.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 14, 10, 12, 20, 22, 18, 26, 28, 30, 18, 34, 24, 38, 30, 42, 44, 46, 30, 30, 52, 30, 42, 58, 60, 62, 36, 66, 68, 70, 42, 74, 76, 78, 50, 82, 84, 86, 66, 60, 92, 94, 54, 56, 60, 102, 78, 106, 60, 110, 70, 114, 116, 118, 90, 122, 124, 84
Offset: 1

Views

Author

Peter Kagey, Feb 07 2015

Keywords

Comments

A073353(n) <= a(n) <= 2*n. Any prime that divides n must also divide a(n), and because n divides (2*n)^4.
a(n) = 2*n iff n is squarefree (A005117). - Robert Israel, Feb 08 2015

Examples

			a(16) = 18 because 16 divides 18^4, but 16 does not divide 17^4.

Links

Crossrefs

Cf. A005117 (squarefree).
Cf. A073353 (similar, with k^n).
Cf. A254732 (similar, with k^2), A254733 (similar, with k^3).

Programs

  • Maple
    f:= proc(n) local k;
     for k from n+1 do if (k^4/n)::integer then return k fi od:
end proc:
seq(f(n), n=1..100); # Robert Israel, Feb 08 2015
  • Mathematica
    lk[n_]:=Module[{k=n+1},While[PowerMod[k,4,n]!=0,k++];k]; Array[lk,70] (* Harvey P. Dale, Nov 22 2015 *)
  • PARI
    a(n)=for(k=n+1,2*n,if(k^4%n==0,return(k)))
vector(100,n,a(n)) \\ Derek Orr Feb 07 2015
  • Python
    def A254734(n):
    k = n + 1
    while pow(k, 4, n):
        k += 1
    return k # Chai Wah Wu, Feb 15 2015
  • Ruby
    def a(n)
  (n+1..2*n).find { |k| k**4 % n == 0 }
end

A073354 Binomial coefficient ( n, squarefree kernel(n) ).

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 28, 84, 1, 1, 924, 1, 1, 1, 120, 1, 18564, 1, 184756, 1, 1, 1, 134596, 53130, 1, 2925, 40116600, 1, 1, 1, 496, 1, 1, 1, 1947792, 1, 1, 1, 847660528, 1, 1, 1, 2104098963720, 344867425584, 1, 1, 12271512, 85900584, 10272278170, 1
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 29 2002

Keywords

Comments

a(n)=1 iff n is squarefree.

Links

Crossrefs

Cf. A007947, A005117, A073353, A066503, A064549, A003557.

Programs

  • Maple
    f:= proc(n) local k;
  k:= convert(numtheory:-factorset(n),`*`);
  binomial(n,k)
end proc:
map(f, [$1..60]); # Robert Israel, May 07 2021
  • Mathematica
    a[n_] := Binomial[n, Times @@ FactorInteger[n][[All, 1]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)

Formula

a(n) = binomial(n, A007947(n)).

A075656 n + product of prime factors of n.

Original entry on oeis.org

1, 4, 6, 6, 10, 12, 14, 10, 12, 20, 22, 18, 26, 28, 30, 18, 34, 24, 38, 30, 42, 44, 46, 30, 30, 52, 30, 42, 58, 60, 62, 34, 66, 68, 70, 42, 74, 76, 78, 50, 82, 84, 86, 66, 60, 92, 94, 54, 56, 60, 102, 78, 106, 60, 110, 70, 114, 116, 118, 90, 122, 124, 84, 66, 130, 132
Offset: 1

Views

Author

Joseph L. Pe, Oct 12 2002

Keywords

Comments

Prime factors taken without multiplicity. - Harvey P. Dale, Feb 27 2021

Examples

			6 + product of prime factors of 6 = 6 + 2 * 3 = 12, so a(6) = 12.

Links

Programs

  • Mathematica
    Flatten[Append[{1}, Table[n + Apply[Times, Transpose[FactorInteger[n]][[1]]], {n, 2, 100}]]]
Join[{1},Table[n+Times@@FactorInteger[n][[All,1]],{n,2,70}]] (* Harvey P. Dale, Feb 27 2021 *)

Formula

a(n)=A073353(n), n>1. [From R. J. Mathar, Sep 23 2008]

A272327 Table read by antidiagonals: T(n, k) is the least i > n such that n divides i^k (n > 0, k > 0).

Original entry on oeis.org

2, 4, 2, 6, 4, 2, 8, 6, 4, 2, 10, 6, 6, 4, 2, 12, 10, 6, 6, 4, 2, 14, 12, 10, 6, 6, 4, 2, 16, 14, 12, 10, 6, 6, 4, 2, 18, 12, 14, 12, 10, 6, 6, 4, 2, 20, 12, 10, 14, 12, 10, 6, 6, 4, 2, 22, 20, 12, 10, 14, 12, 10, 6, 6, 4, 2, 24, 22, 20, 12, 10, 14, 12, 10, 6
Offset: 1

Views

Author

Peter Kagey, Apr 25 2016

Keywords

Comments

T(n, k) = 2*n for squarefree n.

Examples

			a(1) = T(1, 1) = 2  because 1 divides 2^1
a(2) = T(2, 1) = 4  because 2 divides 4^1
a(3) = T(1, 2) = 2  because 1 divides 2^2
a(4) = T(3, 1) = 6  because 3 divides 6^1
a(5) = T(2, 2) = 4  because 2 divides 4^2
a(6) = T(1, 3) = 2  because 1 divides 2^3
a(7) = T(4, 1) = 8  because 4 divides 8^1
a(8) = T(3, 2) = 6  because 3 divides 6^2
a(9) = T(2, 3) = 4  because 2 divides 4^3
a(10) = T(1, 4) = 2 because 1 divides 2^4
Triangle begins:
   2  2 2 2 2 2
   4  4 4 4 4
   6  6 6 6
   8  6 6
  10 10
  12

Links

Crossrefs

Cf. A254732 (second column), A254733 (third column), A254734 (fourth column), A073353 (main diagonal).

Programs

  • Mathematica
    Table[Function[m, SelectFirst[Range[m + 1, 10^3], Divisible[#^k, m] &]][n - k + 1], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Apr 25 2016, Version 10 *)

A075655 Numbers n such that n + product of prime factors of n = (n+1) + product of prime factors of (n+1).

Original entry on oeis.org

3, 24, 1088, 4224, 16640, 66048, 16785408, 67125248, 268468224
Offset: 1

Views

Author

Joseph L. Pe, Oct 12 2002

Keywords

Comments

Numbers n such that A073353(n) = A073353(n+1). - Michel Marcus, Aug 23 2019
No more terms below 10^10. - Amiram Eldar, Aug 23 2019

Examples

			24 + product of prime factors of 24 = 24 + 2 * 3 = 30; 25 + product of prime factors of 25 = 25 + 5 = 30; hence 24 belongs to the sequence.

Crossrefs

Cf. A007947, A073353, A081083, A081084.

Programs

  • Mathematica
    s[n_] := n + Apply[Times, Transpose[FactorInteger[n]][[1]]]; Select[Range[2, 10^5], s[ # ] == s[ # + 1] &]
Flatten[Position[Partition[Table[n+Times@@Transpose[FactorInteger[ n]] [[1]],{n,2,100000}],2,1],?(#[[1]]==#[[2]]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Apr 20 2014 *)
rad[n_] := Times @@ First /@ FactorInteger[n]; r1 = 1; s = {}; Do[r2 = rad[n]; If[r1 - r2 == 1, AppendTo[s, n-1]]; r1 = r2, {n, 2, 10^5}]; s (* Amiram Eldar, Aug 23 2019 *)

Extensions

a(7)-a(9) from Amiram Eldar, Aug 23 2019

A124676 a(n) = largest integer < n such that the positive integers <= a(n) and coprime to a(n) are also coprime to n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 6, 6, 4, 10, 6, 12, 6, 3, 14, 16, 12, 18, 10, 6, 10, 22, 18, 20, 12, 24, 14, 28, 2, 30, 30, 9, 16, 5, 30, 36, 18, 12, 30, 40, 6, 42, 22, 30, 22, 46, 42, 42, 40, 15, 26, 52, 48, 10, 42, 18, 28, 58, 30, 60, 30, 42, 62, 10, 6, 66, 34, 21, 4, 70, 66, 72, 36, 60, 38, 7, 12
Offset: 2

Views

Author

Leroy Quet, Dec 24 2006

Keywords

Examples

			The positive integers k, k <= 6, where the positive integers <= k and coprime to k are also coprime to 6, are 1,2,6. So a(6) = the largest of these < 6, which is 2.

Crossrefs

Cf. A126260, A073353.

Programs

  • Mathematica
    f[n_] := Select[Range[n], GCD[ #, n] == 1 &];g[n_] := Select[Range[n - 1], Times @@ GCD[f[ # ], n] == 1 &];Max /@ Table[g[n], {n, 2, 80}] (* Ray Chandler, Dec 24 2006 *)

Extensions

Extended by Ray Chandler, Dec 24 2006
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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