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 1-5 of 5 results.

A080737 a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.

Original entry on oeis.org

0, 0, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 6, 8, 16, 6, 18, 6, 8, 10, 22, 6, 20, 12, 18, 8, 28, 6, 30, 16, 12, 16, 10, 8, 36, 18, 14, 8, 40, 8, 42, 12, 10, 22, 46, 10, 42, 20, 18, 14, 52, 18, 14, 10, 20, 28, 58, 8, 60, 30, 12, 32, 16, 12, 66, 18, 24, 10, 70, 10, 72, 36, 22, 20, 16, 14
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2003

Keywords

Crossrefs

See A152455 for another version.

Programs

  • Haskell
    a080737 n = a080737_list !! (n-1)
    a080737_list = 0 : (map f [2..]) where
    f n | mod n 4 == 2 = a080737 $ div n 2
    | otherwise = a067240 n
    -- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012
  • Mathematica
    a[1] = a[2] = 0; a[p_?PrimeQ] := a[p] = p-1; a[n_] := a[n] = If[Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[a /@ (fi[[All, 1]]^fi[[All, 2]])]]; Table[a[n], {n, 1, 78}] (* Jean-François Alcover, Jun 20 2012 *)
  • PARI
    for(n=1,78,k=0; if(n>1,f=factor(n); k=sum(j=1,matsize(f)[1],eulerphi(f[j,1]^f[j,2])); if(f[1,1]==2&&f[1,2]==1,k--)); print1(k,",")) \\ Klaus Brockhaus, Mar 10 2003
    

Formula

For n > 2, a(2^r) = 2^(r-1) with r>1, a(p^r) = phi(p^r) with p > 2 prime, r >= 1, where phi is Euler's function A000010; in general if a(Product p_i^e_i) = Sum a(p_i^e_i).

Extensions

More terms from Klaus Brockhaus, Mar 10 2003

A080738 Array read by rows in which 0th row is {1,2}; for n>0, n-th row gives finite orders of 2n X 2n integer matrices that are not orders of 2n-1 X 2n-1 integer matrices.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 10, 12, 7, 9, 14, 15, 18, 20, 24, 30, 16, 21, 28, 36, 40, 42, 60, 11, 22, 35, 45, 48, 56, 70, 72, 84, 90, 120, 13, 26, 33, 44, 63, 66, 80, 105, 126, 140, 168, 180, 210, 39, 52, 55, 78, 88, 110, 112, 132, 144, 240, 252, 280, 360, 420, 17, 32, 34, 65, 77
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2003

Keywords

Comments

A080739 gives number of elements in n-th row.
If k appears in row n, then k-fold rotational symmetry is compatible with some 2n- (or higher) dimensional crystallographic symmetry. - Andrey Zabolotskiy, Jul 08 2017
The set of finite orders of n X n integer matrices = {m : A080737(m) <= n}. This set is also {a(i) : 1<=i <= Sum_{0<=j<=n/2} A080739(j)}. - Günter Rote, Sep 18 2023

Examples

			The array begins:
  1, 2;
  3, 4,  6;
  5, 8, 10, 12;
  7, 9, 14, 15, 18, 20, 24, 30;
  ...
		

Crossrefs

Programs

  • Haskell
    import Data.Map (singleton, deleteFindMin, insertWith)
    a080738 n k = a080738_tabf !! n !! k
    a080738_row n = a080738_tabf !! n
    a080738_tabf = f 3 (drop 2 a080737_list) 3 (singleton 0 [2,1]) where
       f i xs'@(x:xs) till m
         | i > till  = (reverse row) : f i xs' (3 * head row) m'
         | otherwise = f (i + 1) xs till (insertWith (++) (div x 2) [i] m)
         where ((_,row),m')  = deleteFindMin m
    -- Reinhard Zumkeller, Jun 13 2012
  • Mathematica
    a080737[1] = a080737[2] = 0; a080737[p_?PrimeQ] := a080737[p] = p-1; a080737[n_] := a080737[n] = If[ Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[ a080737 /@ (fi[[All, 1]]^fi[[All, 2]])]]; orders = Table[{n, a080737[n]}, {n, 1, 420}]; row[0] = {1, 2};row[n_] := Select[ orders, 2n-1 <= #[[2]] <= 2n & ][[All, 1]]; A080738 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Jun 20 2012 *)

Extensions

More terms from Vladeta Jovovic, Mar 09 2003

A080739 Number of elements in n-th row of A080738.

Original entry on oeis.org

2, 3, 4, 8, 7, 11, 13, 14, 20, 26, 30, 42, 43, 49, 64, 65, 82, 95, 107, 129, 143, 156, 186, 203, 235, 267, 295, 334, 372, 415, 466, 514, 581, 637, 707, 783, 859, 960, 1055, 1152, 1281, 1390, 1522, 1679, 1824, 2011, 2190, 2377, 2595, 2816, 3059, 3342, 3616, 3940
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2003

Keywords

Crossrefs

Programs

  • Haskell
    a080739 n k = a080739_list !! n
    a080739_list = map length a080738_tabf  -- Reinhard Zumkeller, Jun 13 2012

Extensions

More terms from Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003

A080742 Largest element of n-th row of A080738.

Original entry on oeis.org

2, 6, 12, 30, 60, 120, 210, 420, 840, 1260, 2520, 2310, 5040, 9240, 13860, 27720, 32760, 55440, 65520, 120120, 180180, 360360, 235620, 720720, 556920, 942480, 1113840, 1531530
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2003

Keywords

Crossrefs

Programs

  • Haskell
    a080742 n k = a080742_list !! n
    a080742_list = map last a080738_tabf  -- Reinhard Zumkeller, Jun 13 2012
  • Mathematica
    a080737[1] = a080737[2] = 0; a080737[p_?PrimeQ] := a080737[p] = p - 1; a080737[n_] := a080737[n] = If[Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[a080737 /@ (fi[[All, 1]]^fi[[All, 2]])]]; orders = Table[{n, a080737[n]}, {n, 1, 2*10^6}]; row[0] = {1, 2}; row[n_] := Select[orders, 2*n - 1 <= #[[2]] <= 2*n &][[All, 1]]; Table[ row[n] // Last, {n, 0, 27}] (* Jean-François Alcover, Jul 24 2013 *)

Extensions

Corrected and extended by Vladeta Jovovic, Mar 09 2003

A080741 Smallest element of n-th row of A080738.

Original entry on oeis.org

1, 3, 5, 7, 16, 11, 13, 39, 17, 19, 25, 23, 69, 115, 29, 31, 64, 155, 37, 111, 41, 43, 129, 47, 141, 235, 53, 81, 265, 59, 61, 183, 128, 67, 201, 71, 73, 219, 365, 79, 237, 83, 249, 415, 89, 267, 445, 623, 97, 291, 101, 103, 309, 107, 109, 121, 113, 339, 565, 791, 1417
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2003

Keywords

Crossrefs

Programs

  • Haskell
    a080741 n k = a080741_list !! n
    a080741_list = map head a080738_tabf  -- Reinhard Zumkeller, Jun 13 2012

Extensions

More terms from Vladeta Jovovic, Mar 09 2003
Showing 1-5 of 5 results.