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.

A299143 a(n) is the least k > n such that gcd(k,n) > 1 and gcd(k+1,n+1) > 1.

Original entry on oeis.org

8, 9, 14, 15, 20, 21, 14, 15, 32, 33, 38, 39, 20, 21, 50, 51, 56, 57, 26, 27, 68, 69, 34, 35, 32, 33, 86, 87, 92, 93, 38, 39, 44, 45, 110, 111, 44, 45, 122, 123, 128, 129, 50, 51, 140, 141, 62, 63, 56, 57, 158, 159, 64, 65, 62, 63, 176, 177, 182, 183, 68, 69
Offset: 2

Views

Author

Alex Ratushnyak, Feb 03 2018

Keywords

Examples

			8 is the least k>2 such that gcd(8,2)>1 and gcd(9,3)>1. So a(2)=8.
15 is the least k>9 such that gcd(15,9)>1 and gcd(16,10)>1. Therefore a(9)=15.

Links

Crossrefs

Cf. A172170.
Cf. A061228 or A159475 (when simply gcd(k,n) > 1).

Programs

  • Maple
    f:= proc(n) local k;
      for k from n+1 do if igcd(k,n)>1 and igcd(k+1,n+1)>1 then return k fi od
end proc:
map(f, [$2..100]); # Robert Israel, Mar 08 2018
  • Mathematica
    Array[Block[{k = # + 1}, While[Or[CoprimeQ[#, k], CoprimeQ[# + 1, k + 1]], k++]; k] &, 62, 2] (* Michael De Vlieger, Feb 03 2018 *)
  • PARI
    a(n) = for (k=n+1, oo, if (gcd(n,k)>1 && gcd(n+1, k+1)>1, return (k))) \\ Rémy Sigrist, Feb 04 2018

Formula

From Rémy Sigrist, Feb 04 2018: (Start)
a(p) = 3 * p for any odd prime p.
a(2*k + 1) = a(2*k) + 1 for any k > 0.
a(n) = n + 2*A172170(n + 1) for any n > 1.
(End)

A322292 a(n) = Max_{c composite, c < n} (c + least prime factor of c).

Original entry on oeis.org

6, 6, 8, 8, 10, 12, 12, 12, 14, 14, 16, 18, 18, 18, 20, 20, 22, 24, 24, 24, 26, 30, 30, 30, 30, 30, 32, 32, 34, 36, 36, 40, 40, 40, 40, 42, 42, 42, 44, 44, 46, 48, 48, 48, 50, 56, 56, 56, 56, 56, 56, 60, 60, 60, 60, 60, 62, 62, 64, 66, 66, 70, 70, 70, 70, 72, 72, 72
Offset: 5

Views

Author

Michel Marcus, Dec 02 2018

Keywords

Comments

a(n) is only defined for n >= 5, because for n < 5, the condition {c composite, c < n} results in the empty set.

Examples

			a(5) = 6 because the largest composite c < n = 5 is 4, which has the largest prime factor 2. Hence a(5) = 4 + 2 = 6. - _David A. Corneth_, Dec 03 2018

Links

Crossrefs

Cf. A061228, A159475, A322293.

Programs

  • Maple
    N:= 100: # to get a(5)..a(N)
V:= Vector(N):
V[5]:= 6;
for n from 6 to N do
  if isprime(n-1) then V[n]:= V[n-1]
  else V[n]:= max(V[n-1],n-1+min(numtheory:-factorset(n-1)))
  fi
od:
convert(V[5..N],list); # Robert Israel, Dec 03 2018
  • Mathematica
    a[n_] := Module[{smax = 0}, Do[If[CompositeQ[m],  smax = Max[smax, m + FactorInteger[m][[1, 1]]]], {m, 2, n-1}]; smax]; Array[a, 100, 5] (* Amiram Eldar, Dec 02 2018 *)
  • PARI
    a(n) = {my(smax = 0); for(m=2, n-1, if (!isprime(m), smax = max(smax, m + factor(m)[1, 1]); )); smax; }

A322293 Integers k such that A322292(k) <= k.

Original entry on oeis.org

6, 8, 12, 14, 18, 20, 24, 30, 32, 42, 44, 48, 60, 62, 72, 74, 84, 90, 102, 104, 108, 110, 114, 132, 140, 168, 182, 198, 200, 234, 240, 242, 270, 272, 282, 284, 312, 314, 318, 354, 360, 390, 420, 422, 434, 462, 464, 468, 510, 572, 648, 660, 662, 762, 840, 884, 888, 942, 1064
Offset: 1

Views

Author

Michel Marcus, Dec 02 2018

Keywords

Comments

Erdos conjectures that this sequence is finite.

Links

Crossrefs

Cf. A061228, A159475, A322292.

Programs

  • Maple
    N:= 10^6: # to get all terms <= N
Res:= 6: v:= 6:
for n from 7 to N do
  if not isprime(n-1) then v:= max(v, n-1 + min(numtheory:-factorset(n-1))) fi;
if v <= n then Res:= Res, n fi;
od:
Res; # Robert Israel, Dec 03 2018
  • Mathematica
    f[n_] := Module[{smax = 0}, Do[If[CompositeQ[m],  smax = Max[smax, m + FactorInteger[m][[1, 1]]]], {m, 2, n-1}]; smax];aQ[n_] := f[n]<=n; Select[Range[6, 1000], aQ] (* Amiram Eldar, Dec 02 2018 *)
  • PARI
    f(n) = {my(smax = 0); forcomposite(m=1, n-1, smax = max(smax, m + factor(m)[1,1]);); smax;} \\ A322292
isok(n) = f(n) <= n;
Showing 1-3 of 3 results.

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

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