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.

A124738 Irregular table where the n-th row consists of those positive integers which are coprime to both n and n+1 and which are <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 1, 3, 5, 1, 5, 7, 1, 7, 1, 3, 7, 9, 1, 5, 7, 1, 5, 7, 11, 1, 3, 5, 9, 11, 1, 11, 13, 1, 7, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 5, 7, 11, 13, 1, 5, 7, 11, 13, 17, 1, 3, 7, 9, 11, 13, 17, 1, 11, 13, 17, 19, 1, 5, 13, 17, 19, 1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 1, 5, 7, 11, 13
Offset: 1

Views

Author

Leroy Quet, Nov 06 2006

Keywords

Comments

The n-th row has A057475(n) terms.

Examples

			The positive integers which are coprime to 8 and which are <= 8 are 1,3,5,7. The integers which are coprime to 9 and which are <= 9 are 1, The integers in both these sequences (1,5,7) make up the row of A124738.

Crossrefs

Cf. A057475, A124739, A124740, A124741.

Programs

  • Mathematica
    f[n_] := Select[Range[n], GCD[n, # ] == GCD[n + 1, # ] == 1 &]; Flatten[Table[f[n], {n, 23}]] (* Ray Chandler, Nov 10 2006 *)

Extensions

Extended by Ray Chandler, Nov 10 2006

A124740 a(n) = product of those positive integers which are coprime to both n and n+1 and which are <= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 15, 35, 7, 189, 35, 385, 1485, 143, 1001, 2027025, 5005, 85085, 459459, 46189, 20995, 1249937325, 1616615, 7436429, 324342711, 71504125, 132793375, 1452095555625, 7436429, 215656441, 6190283353629375, 75969882625
Offset: 1

Views

Author

Leroy Quet, Nov 06 2006

Keywords

Examples

			The positive integers which are coprime to 8 and which are <= 8 are 1,3,5,7. The positive integers which are coprime to 9 and which are <= 9 are 1, 2,4,5,7,8. The integers in both these sequences (1,5,7) are multiplied to get a(8) = 35.

Crossrefs

Cf. A057475, A124738, A124739, A124741.

Programs

  • Mathematica
    f[n_] := Times @@ Select[Range[n], GCD[n, # ] == GCD[n + 1, # ] == 1 &];Table[f[n], {n, 33}] (* Ray Chandler, Nov 10 2006 *)

Extensions

Extended by Ray Chandler, Nov 10 2006

A124741 a(n) = largest of those positive integers which are coprime to both n and n+1 and which are <= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 5, 7, 7, 9, 7, 11, 11, 13, 13, 15, 13, 17, 17, 19, 19, 21, 19, 23, 23, 25, 25, 27, 23, 29, 29, 31, 31, 33, 31, 35, 35, 37, 37, 39, 37, 41, 41, 43, 43, 45, 43, 47, 47, 49, 49, 51, 49, 53, 53, 55, 55, 57, 53, 59, 59, 61, 61, 63, 61, 65, 65, 67, 67, 69, 67, 71, 71
Offset: 1

Views

Author

Leroy Quet, Nov 06 2006

Keywords

Examples

			The positive integers which are coprime to 8 and which are <= 8 are 1,3,5,7. The positive integers which are coprime to 9 and which are <= 9 are 1, 2,4,5,7,8. So a(8) = 7, which is the largest of those integers in both these sequences (1,5,7).

Links

Crossrefs

Cf. A057475, A124738, A124739, A124740.

Programs

  • Mathematica
    f[n_] := Last @ Select[Range[n], GCD[n, # ] == GCD[n + 1, # ] == 1 &];Table[f[n], {n, 75}] (* Ray Chandler, Nov 10 2006 *)
Join[{1},Table[Max[Intersection[Select[Range[n-1],CoprimeQ[ #,n]&],Select[ Range[n-1],CoprimeQ[#,n+1]&]]],{n,2,80}]] (* Harvey P. Dale, Jul 08 2018 *)

Extensions

Extended by Ray Chandler, Nov 10 2006

A124739 a(n) = sum of those positive integers which are coprime to both n and n+1 and which are <= n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 9, 13, 8, 20, 13, 24, 29, 25, 32, 64, 37, 54, 61, 61, 55, 110, 73, 91, 111, 108, 114, 168, 91, 120, 225, 170, 153, 199, 144, 216, 305, 221, 175, 320, 211, 252, 397, 261, 249, 506, 337, 342, 423, 351, 403, 624, 433, 410, 483, 431, 493, 812, 421, 480
Offset: 1

Views

Author

Leroy Quet, Nov 06 2006

Keywords

Examples

			The positive integers which are coprime to 8 and which are <= 8 are 1,3,5,7. The integers which are coprime to 9 and which are <= 9 are 1, The integers in both these sequences (1,5,7) are added get a(8) = 13.

Crossrefs

Cf. A057475, A124738, A124740, A124741.

Programs

  • Mathematica
    f[n_] := Plus @@ Select[Range[n], GCD[n, # ] == GCD[n + 1, # ] == 1 &];Table[f[n], {n, 60}] (* Ray Chandler, Nov 10 2006 *)

Extensions

Extended by Ray Chandler, Nov 10 2006

A186230 Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of positive integers j

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 2, 4, 2, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 1, 0, 1, 3, 0, 4, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 5, 0
Offset: 1

Views

Author

Alois P. Heinz, Feb 15 2011

Keywords

Comments

T(n,k) = A000010(k) if n is prime and 1

Examples

			T(n,1) = 0 because no positive integer j<1 can be found.
T(n,k) = 0 if GCD(n,k)>1.
T(7,5) = 4 because for j in {1,2,3,4} all conditions are satisfied.
Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 1, 2, 2, 0;
  0, 0, 0, 0, 1, 0;
  0, 1, 2, 2, 4, 2, 0;

Links

Crossrefs

Row sums give: A185953. Column k=2 gives: A000035 for n>1. Lower diagonal gives: A057475(n-1) for n>2. Cf. A000010, A000040, A003989.

Programs

  • Maple
    with(numtheory):
T:= proc(n,k) local c, i, j, m;
      if k=1 or igcd(n, k)>1 then 0
    elif isprime(n) then phi(k)
    else m:= n*k;
         i:= igcd(m, 2);
         c:= 0;
         for j to k-1 by i do
           if igcd(m, j)=1 then c:= c+1 fi
         od; c
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..20);
  • Mathematica
    t[n_, k_] := Module[{c, i, j, m}, If[ k == 1 || GCD[n, k] > 1, 0, If[PrimeQ[n], EulerPhi[k], m = n*k; i = GCD[m, 2]; c = 0; For[j = 1, j <= k-1, j = j+i, If[GCD[m, j] == 1, c = c+1]]; c]]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

Formula

T(n,k) = |{ j : 1 <= j < k and GCD(n,k) = GCD(n,j) = GCD(k,j) = 1 }|.

A118854 Numbers m such that m-1 and m have the same number of common totatives as m and m+1 have.

Original entry on oeis.org

2, 3, 8, 21, 24, 27, 45, 75, 93, 105, 117, 123, 147, 165, 213, 309, 315, 333, 357, 387, 453, 525, 555, 573, 627, 636, 693, 717, 729, 765, 795, 843, 915, 933, 957, 1005, 1083, 1125, 1173, 1227, 1323, 1347, 1437, 1467, 1515, 1563, 1575, 1677, 1725, 1755, 1773
Offset: 1

Views

Author

Reinhard Zumkeller, May 02 2006

Keywords

Comments

A057475(a(n)-1) = A057475(a(n));
it seems that even values are very rare, see A118855.

Examples

			n = 21, the sets of totatives for 21-1, 21 and 21+1:
T(20) = {1,3,7,9,11,13,17,19},
T(21) = {1,2,4,5,8,10,11,13,16,17,19,20},
T(22) = {1,3,5,7,9,13,15,17,19,21},
A057475(20) = #intersect(T(20),T(21)) = #{1,11,13,17,19} = 5,
A057475(20) = #intersect(T(21),T(22)) = #{1,5,13,17,19} = 5,
therefore 21 is a term.

Links

  • Eric Weisstein's World of Mathematics, Totative

A118855 Even terms in A118854.

Original entry on oeis.org

2, 8, 24, 636, 12318, 13446, 32396, 46206, 133788, 162926, 181428, 359906, 439098, 453168, 485238, 508200, 739024, 840852, 1007580, 1410120, 1577066, 2056076, 2126134, 2422566, 2512406, 3275832, 3307848, 3457758, 4481014, 4698784
Offset: 1

Views

Author

Reinhard Zumkeller, May 02 2006

Keywords

Crossrefs

Cf. A057475, A118854.

Extensions

a(9)-a(30) from Donovan Johnson, Dec 26 2010

A078639 Number of nonisomorphic graphic matroids (by rank or number of vertices).

Original entry on oeis.org

1, 2, 5, 16, 73, 533, 7303
Offset: 1

Views

Author

Gordon Royle, Oct 06 2008

Keywords

Comments

Could A002632 be an erroneous version of this sequence?
The old entry with this sequence number was a duplicate of A057475.
Showing 1-8 of 8 results.

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