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 1-4 of 4 results.

A138389 Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 20, 21, 23, 24, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199
Offset: 1

Views

Author

Vladimir Shevelev, May 08 2008

Keywords

Comments

Note that every i not exceeding n/2 for which (n,i)=1 divides binomial(n-i-1,i-1). For n>33, a(n) is either prime or square of a prime or a product of twin primes. For a proof, see link of V. Shevelev.
Numbers n such that A178105(n) = 0. - Michel Marcus, Feb 07 2016

Links

Crossrefs

Cf. A000040, A001248, A077800, A037074, A178105.

Programs

  • Mathematica
    Select[Range@ 200, Function[n, NoneTrue[Select[Range@ Floor[n/2], ! CoprimeQ[#, n] &], Divisible[Binomial[n - # - 1, # - 1], #] &]]] (* Michael De Vlieger, Feb 07 2016, Version 10 *)
  • PARI
    isok(n) = {my(md = -1); for (d=2, n\2, if (((binomial(n-d-1,d-1) % d) == 0) && (gcd(n, d) > 1), if (md == -1, md = d, md = min(d, md)));); (md == -1);} \\ Michel Marcus, Feb 07 2016

A178101 Cardinality of the set of d, 2<=d<=n/2, which divide binomial(n-d-1,d-1) and are not coprime to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 2, 0, 3, 0, 4, 0, 2, 0, 3, 1, 2, 0, 2, 0, 2, 3, 5, 0, 5, 0, 6, 3, 3, 0, 6, 1, 3, 3, 8, 0, 5, 0, 11, 3, 8, 1, 8, 0, 5, 3, 7, 0, 6, 0, 8, 4, 5, 1, 7, 0, 7, 5, 10, 0, 4, 1, 9, 3, 6, 0, 10, 2, 8, 4, 15, 2, 10, 0, 16, 6, 10
Offset: 1

Views

Author

Vladimir Shevelev, May 20 2010

Keywords

Comments

Note that every d>1 divides binomial(n-d-1,d-1), if gcd(n,d)=1.
Numbers n with cardinality 0 are in A138389, with cardinatly 1 in A178071, with cardinality 2 in A178098 and with cardinality 3 in A178099.

Links

Crossrefs

Cf. A138389, A178071, A178098, A179099, A178105.

Programs

  • Maple
    A178101 := proc(n) local dvs,d ; dvs := {} ; for d from 1 to n/2 do if gcd(n,d) > 1 and d in numtheory[divisors]( binomial(n-d-1,d-1)) then dvs := dvs union {d} ; end if; end do: nops(dvs) end proc: # R. J. Mathar, May 28 2010
  • Mathematica
    a[n_] := Sum[Boole[Divisible[Binomial[n-d-1, d-1], d] && !CoprimeQ[d, n]], {d, 2, n/2}];
Array[a, 100] (* Jean-François Alcover, Nov 17 2017 *)
  • PARI
    a(n) = sum(d=2, n\2, (gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0)); \\ Michel Marcus, Feb 17 2016

Extensions

a(54), a(68), a(70), a(72), a(78) etc corrected by R. J. Mathar, May 28 2010

A178109 The maximum number d, 2 <= d <= n/2, which divides binomial(n-d-1,d-1) and is not coprime to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 0, 6, 0, 0, 0, 4, 0, 0, 0, 8, 6, 10, 0, 8, 0, 14, 0, 14, 0, 15, 0, 14, 9, 14, 0, 9, 0, 10, 15, 20, 0, 22, 0, 22, 21, 18, 0, 21, 20, 22, 24, 24, 0, 26, 0, 28, 24, 30, 10, 28, 0, 30, 24, 26, 0, 33, 0, 30, 20, 30, 21, 28, 0, 38, 33, 38, 0, 28, 20, 36
Offset: 1

Views

Author

Vladimir Shevelev, May 20 2010

Keywords

Comments

If no such divisors d exist, a(n)=0.

Links

Crossrefs

Cf. A138389, A178071, A178098 - A178101, A178105

Programs

  • Maple
    A178109 := proc(n) local dvs,d ; dvs := {} ; for d from 1 to n/2 do if gcd(n,d) > 1 and d in numtheory[divisors]( binomial(n-d-1,d-1)) then dvs := dvs union {d} ; end if; end do:
if nops(dvs) = 0 then 0; else max(op(dvs)) ; end if; end proc:
seq(A178109(n),n=1..90) ; # R. J. Mathar, May 28 2010
# Alternative:
f:= proc(n) local d;
  for d from floor(n/2) to 2 by -1 do
     if igcd(d,n) > 1 and binomial(n-d-1,d-1) mod d = 0 then return d fi
  od;
  0
end proc:
map(f, [$1..100]); # Robert Israel, Jan 15 2019
  • Mathematica
    a[n_] := If[n==1, 0, Module[{d=Floor[n/2]}, While[d>1 && (GCD[n, d]==1 || !Divisible[Binomial[n-d-1,d-1], d]), d--]; If[d==1, d=0]; d]]; Array[a, 100] (* Amiram Eldar, Dec 04 2018 *)

Extensions

a(39), a(54) and a(70) corrected by R. J. Mathar, May 28 2010

A178110 Consider the set of divisors d of binomial(n-d-1,d-1) where gcd(n,d)>1 and 1

Original entry on oeis.org

16, 18, 26, 27, 32, 34, 40, 45, 50, 56, 58, 63, 64, 72, 74, 80, 81, 82, 88, 90, 98, 99, 104, 106, 112, 117, 122, 128, 130, 135, 136, 144, 146, 152, 153, 154, 160, 162, 170, 171, 176, 178, 184, 189, 194, 200, 202, 207, 208, 216, 218, 224, 225, 226, 232, 234, 242
Offset: 1

Views

Author

Vladimir Shevelev, May 20 2010

Keywords

Examples

			The set for n =14 is {4}, which does not admit 14 into the sequence.
The set for n =16 is {6}, which adds 16 to the sequence.
The set for n = 38 is {4,12,14}, which does not admit 38 into the sequence.

Links

Crossrefs

Cf. A138389, A178071, A178098, A178099, A178100, A178101, A178105, A178109.

Programs

  • Maple
    isA178110 := proc(n) local dvs, d ; dvs := {} ; for d from 1 to n/2 do if gcd(n, d) > 1 and d in numtheory[divisors]( binomial(n-d-1, d-1)) then dvs := dvs union {d} ; end if; end do: return (min(op(dvs)) = 6) ; end proc:
for n from 1 to 100 do if isA178110(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Aug 20 2010
  • Mathematica
    bQ[n_] := Module[{B={}}, Do[If[GCD[i,n]>1 && Divisible[Binomial[n-i-1,i-1], i], AppendTo[B,i]], {i, 2, Floor[n/2]}]; Min[B]==6]; Select[Range[250], bQ] (* Amiram Eldar, Jan 20 2019 *)

Extensions

39 removed and 82 added by R. J. Mathar, Aug 20 2010
More terms from Amiram Eldar, Jan 20 2019
Showing 1-4 of 4 results.

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