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.

A083505 Erroneous version of A027428.

Original entry on oeis.org

1, 3, 6, 10, 14
Offset: 2

Views

Author

Keywords

A027430 Number of distinct products i*j*k with 1 <= i < j < k <= n.

Original entry on oeis.org

0, 0, 1, 4, 10, 16, 29, 42, 60, 75, 111, 126, 177, 206, 238, 274, 361, 396, 507, 554, 613, 677, 838, 883, 1004, 1092, 1198, 1277, 1529, 1590, 1881, 1998, 2133, 2275, 2432, 2518, 2921, 3096, 3278, 3391, 3884, 4014, 4563, 4750, 4938, 5186, 5840, 5987, 6422, 6652
Offset: 1

Views

Author

Keywords

References

  • Amarnath Murthy, Generalization of partition function introducing Smarandache Factor Partitions, Smarandache Notions Journal, 1-2-3, Vol. 11, 2000.

Crossrefs

Number of terms in row n of A083507.

Programs

  • Haskell
    import Data.List (nub)
    a027430 n = length $ nub [i*j*k | k<-[3..n], j<-[2..k-1], i<-[1..j-1]]
    -- Reinhard Zumkeller, Jan 01 2012
    
  • Mathematica
    nn = 50;
    prod = Table[0, {1 + nn^3}];
    a[1] = 0;
    a[n_] := (Do[prod[[1 + i*j*k]] = 1, {i, 0, n}, {j, i+1, n}, {k, j+1, n}]; Count[Take[prod, 1 + n^3], 1] - 1);
    Array[a, nn] (* Jean-François Alcover, Jul 31 2018, after T. D. Noe *)
  • PARI
    \\ See PARI link. David A. Corneth, Jul 31 2018
    
  • Python
    def A027430(n): return len({i*j*k for i in range(1,n+1) for j in range(1,i) for k in range(1,j)}) # Chai Wah Wu, Oct 16 2023

Formula

a(n) = A027429(n)-1. - T. D. Noe, Jan 16 2007
a(n) <= A000292(n - 2). - David A. Corneth, Jul 31 2018

Extensions

Corrected by David Wasserman, Nov 18 2004

A100435 Number of distinct products i*j*k for 1 <= i <= j < k <= n.

Original entry on oeis.org

0, 1, 4, 9, 18, 26, 44, 57, 76, 93, 135, 153, 212, 245, 282, 317, 414, 452, 575, 624, 690, 759, 935, 986, 1103, 1196, 1297, 1378, 1645, 1716, 2024, 2136, 2279, 2427, 2597, 2687, 3110, 3292, 3483, 3606, 4123, 4262, 4837, 5026, 5227, 5485, 6168, 6318, 6725
Offset: 1

Views

Author

N. J. A. Sloane, Nov 21 2004

Keywords

Crossrefs

Programs

  • Maple
    f:=proc(n) local i,j,k,t1; t1:={}; for i from 1 to n-1 do for j from i to n-1 do for k from j+1 to n do t1:={op(t1),i*j*k}; od: od: od: t1:=convert(t1,list); nops(t1); end;
  • Mathematica
    f[n_] := Length[ Union[ Flatten[ Table[ i*j*k, {i, n}, {j, i, n}, {k, j + 1, n}] ]]]; Table[ f[n], {n, 49}] (* Robert G. Wilson v, Dec 14 2004 *)
  • Python
    def A100435(n): return len({i*j*k for i in range(1,n+1) for j in range(1,i) for k in range(1,j+1)}) # Chai Wah Wu, Oct 16 2023

Extensions

More terms from Robert G. Wilson v, Dec 14 2004

A027427 Number of distinct products ij with 0 <= i < j <= n.

Original entry on oeis.org

0, 1, 2, 4, 7, 11, 14, 20, 25, 32, 37, 47, 52, 64, 71, 79, 88, 104, 112, 130, 140, 151, 162, 184, 193, 211, 224, 240, 253, 281, 292, 322, 338, 355, 372, 391, 404, 440, 459, 479, 494, 534, 550, 592, 612, 632, 655, 701, 718, 753, 775, 801, 824, 876
Offset: 0

Views

Author

Keywords

Crossrefs

Cf. A027430, etc.

Programs

  • Haskell
    import Data.List (nub)
    a027427 n = length $ nub [i*j | j <- [1..n], i <- [0..j-1]]
    -- Reinhard Zumkeller, Jan 01 2012
    
  • Maple
    A027427 := proc(n)
        local L, i, j ;
        L := {};
        for i from 0 to n do
            for j from i+1 to n do
                L := L union {i*j};
            end do:
        end do:
        nops(L);
    end proc:  # R. J. Mathar, Jun 09 2016
  • Mathematica
    a[n_] := Table[i*j, {i, 0, n}, {j, i+1, n}] // Flatten // Union // Length;
    Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 03 2018 *)
  • Python
    def A027427(n): return 1+len({i*j for i in range(1,n+1) for j in range(1,i)}) if n else 0 # Chai Wah Wu, Oct 13 2023

Formula

a(n) = A027428(n)+1. - T. D. Noe, Jan 16 2007

A083506 n-th row of the following triangle contains all distinct numbers that can be obtained as the product of two distinct numbers chosen from 1 to n. (n>1). Sequence contains the triangle read by rows.

Original entry on oeis.org

2, 2, 3, 6, 2, 3, 4, 6, 8, 12, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 20, 24, 30, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 42, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 35, 40, 42, 48, 56, 2, 3, 4, 5, 6, 7, 8, 9
Offset: 2

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 05 2003

Keywords

Examples

			2
2,3,6
2,3,4,6,8,12
2,3,4,5,6,8,10,12,15,20
2,3,4,5,6,8,10,12,15,16,18,20,24,30
2,3,4,5,6,7,8,10,12,...
...
		

References

  • Amarnath Murthy, Generalization of partition function introducing Smarandache Factor Partitions, Smarandache Notions Journal, 1-2-3, Vol. 11, 2000.

Crossrefs

A027428 gives the row lengths.

Extensions

More terms from David Wasserman, Nov 18 2004
Showing 1-5 of 5 results.