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 1-6 of 6 results.

A049791 a(n) = Sum_{k=1..n} T(n,k), array T as in A049790.

Original entry on oeis.org

1, 5, 14, 30, 54, 91, 137, 202, 280, 380, 492, 644, 799, 994, 1212, 1471, 1735, 2071, 2400, 2811, 3232, 3709, 4190, 4804, 5380, 6046, 6739, 7535, 8297, 9246, 10115, 11153, 12184, 13320, 14458, 15839, 17074, 18493, 19931, 21583, 23100, 24942, 26609, 28564, 30517, 32593, 34585, 37048, 39231, 41735, 44187, 46911
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049790, A049792, A049793, A049794, A049795, A049796.

Programs

  • GAP
    List([1..60], n-> Sum([1..n], k-> Sum([1..k], j-> Int(n/Int(k/j)) ))); # G. C. Greubel, Dec 10 2019
  • Magma
    [ &+[(&+[Floor(n/Floor(k/j)): j in [1..k]]): k in [1..n]] n in [1..60]]; // G. C. Greubel, Dec 10 2019
  • Maple
    seq( add(add(floor(n/floor(k/j)), j=1..k), k=1..n), n=1..60); # G. C. Greubel, Dec 10 2019
  • Mathematica
    Table[Sum[Sum[Floor[n/Floor[k/j]], {j, k}], {k, n}], {n, 1, 60}] (* G. C. Greubel, Dec 10 2019 *)
  • PARI
    a(n) = sum(k=1,n, sum(j=1,k, n\(k\j) ));
vector(60, n, a(n)) \\ G. C. Greubel, Dec 10 2019
  • Sage
    [sum(sum(floor(n/floor(k/j)) for j in (1..k)) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Dec 10 2019

Extensions

Terms a(40) onward added by G. C. Greubel, Dec 10 2019

A049793 a(n) = T(n,n-1), array T as in A049790.

Original entry on oeis.org

1, 2, 4, 9, 13, 22, 27, 39, 47, 61, 71, 91, 100, 122, 137, 160, 175, 205, 220, 253, 272, 304, 326, 368, 386, 427, 455, 497, 523, 575, 598, 651, 683, 733, 768, 830, 856, 918, 959, 1021, 1056, 1129, 1162, 1236, 1281, 1347, 1393
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049790, A049791, A049792, A049794, A049795, A049796.

Programs

  • GAP
    a:= function(n)
    if n=2 then return 1;
    else return Sum([1..n-2], j-> Int((n-1)/Int((n-2)/j)) );
    fi; end;
List([2..60], n-> a(n) ); # G. C. Greubel, Dec 10 2019
  • Magma
    [n eq 2 select 1 else (&+[Floor((n-1)/Floor((n-2)/j)): j in [1..n-2]]): n in [2..60]]; // G. C. Greubel, Dec 10 2019
  • Maple
    seq( `if`(n=2, 1, add(floor((n-1)/floor((n-2)/j)), j=1..n-2)), n=2..60); # G. C. Greubel, Dec 10 2019
  • Mathematica
    Table[If[n==2, 1, Sum[Floor[(n-1)/Floor[(n-2)/j]], {j,n-2}]], {n, 2,60}] (* G. C. Greubel, Dec 10 2019 *)
  • PARI
    a(n) = if(n==2, 1, sum(j=1,n-2, (n-1)\((n-2)\j)) );
vector(60, n, a(n+1) ) \\ G. C. Greubel, Dec 10 2019
  • Sage
    def a(n):
    if (n==2): return 1
    else: return sum(floor((n-1)/floor((n-2)/j)) for j in (1..n-2))
[a(n) for n in (2..60)] # G. C. Greubel, Dec 10 2019

A049794 a(n) = T(n,n-2), array T as in A049790.

Original entry on oeis.org

1, 2, 3, 6, 11, 16, 25, 31, 44, 53, 66, 79, 97, 108, 130, 148, 168, 187, 214, 233, 265, 286, 315, 344, 381, 402, 442, 474, 511, 545, 590, 619, 670, 705, 751, 797, 848, 880, 940, 988, 1041, 1087, 1150, 1192, 1263, 1311, 1370, 1431
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049790, A049791, A049792, A049793, A049795, A049796.

Programs

  • GAP
    a:= function(n)
    if n<5 then return n-2;
    else return Sum([1..n-4], j-> Int((n-2)/Int((n-4)/j)) );
    fi; end;
List([3..60], n-> a(n) ); # G. C. Greubel, Dec 10 2019
  • Magma
    [n lt 5 select n-2 else (&+[Floor((n-2)/Floor((n-4)/j)): j in [1..n-4]]): n in [3..60]]; // G. C. Greubel, Dec 10 2019
  • Maple
    seq( `if`(n<5, n-2, add(floor((n-2)/floor((n-4)/j)), j=1..n-4)), n=3..60); # G. C. Greubel, Dec 10 2019
  • Mathematica
    Table[If[n<5, n-2, Sum[Floor[(n-2)/Floor[(n-4)/j]], {j,n-4}]], {n, 3,60}] (* G. C. Greubel, Dec 10 2019 *)
  • PARI
    a(n) = if(n<5, n-2, sum(j=1,n-4, (n-2)\((n-4)\j)) );
vector(60, n, a(n+2) ) \\ G. C. Greubel, Dec 10 2019
  • Sage
    def a(n):
    if (n<5): return n-2
    else: return sum(floor((n-2)/floor((n-4)/j)) for j in (1..n-4))
[a(n) for n in (3..60)] # G. C. Greubel, Dec 10 2019

A049795 a(n) = T(n,n-3), array T as in A049790.

Original entry on oeis.org

1, 2, 3, 4, 7, 14, 18, 29, 35, 49, 57, 74, 84, 105, 115, 140, 155, 180, 195, 228, 244, 278, 296, 332, 356, 397, 416, 460, 487, 534, 559, 612, 637, 691, 722, 779, 814, 872, 901, 968, 1007, 1073, 1107, 1180, 1218, 1292, 1333, 1407
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049790, A049791, A049792, A049793, A049794, A049796.

Programs

  • GAP
    a:= function(n)
    if n<7 then return n-3;
    else return Sum([1..n-6], j-> Int((n-3)/Int((n-6)/j)) );
    fi; end;
List([4..60], n-> a(n) ); # G. C. Greubel, Dec 10 2019
  • Magma
    [n lt 7 select n-3 else (&+[Floor((n-3)/Floor((n-6)/j)): j in [1..n-6]]): n in [4..60]]; // G. C. Greubel, Dec 10 2019
  • Maple
    seq( `if`(n<7, n-3, add(floor((n-3)/floor((n-6)/j)), j=1..n-6)), n=4..60); # G. C. Greubel, Dec 10 2019
  • Mathematica
    Table[If[n<7, n-3, Sum[Floor[(n-3)/Floor[(n-6)/j]], {j,n-6}]], {n,4,60}] (* G. C. Greubel, Dec 10 2019 *)
  • PARI
    a(n) = if(n<7, n-3, sum(j=1,n-6, (n-3)\((n-6)\j)) );
vector(60, n, a(n+3) ) \\ G. C. Greubel, Dec 10 2019
  • Sage
    def a(n):
    if (n<7): return n-3
    else: return sum(floor((n-3)/floor((n-6)/j)) for j in (1..n-6))
[a(n) for n in (4..60)] # G. C. Greubel, Dec 10 2019

A049796 a(n) = T(n,n-4), array T as in A049790.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 16, 22, 32, 39, 53, 64, 79, 91, 113, 124, 147, 166, 188, 208, 238, 256, 288, 313, 344, 371, 411, 434, 473, 509, 548, 580, 629, 657, 709, 749, 796, 837, 893, 928, 987, 1038, 1093, 1136, 1206, 1246, 1314, 1371, 1431
Offset: 5

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049790, A049791, A049792, A049793, A049794, A049795.

Programs

  • GAP
    a:= function(n)
    if n<9 then return n-4;
    else return Sum([1..n-8], j-> Int((n-4)/Int((n-8)/j)) );
    fi; end;
List([5..60], n-> a(n) ); # G. C. Greubel, Dec 10 2019
  • Magma
    [n lt 9 select n-4 else (&+[Floor((n-4)/Floor((n-8)/j)): j in [1..n-8]]): n in [5..60]]; // G. C. Greubel, Dec 10 2019
  • Maple
    seq( `if`(n<9, n-4, add(floor((n-4)/floor((n-8)/j)), j=1..n-8)), n=5..60); # G. C. Greubel, Dec 10 2019
  • Mathematica
    Table[If[n<9, n-4, Sum[Floor[(n-4)/Floor[(n-8)/j]], {j,n-8}]], {n,5,60}] (* G. C. Greubel, Dec 10 2019 *)
  • PARI
    a(n) = if(n<9, n-4, sum(j=1,n-8, (n-4)\((n-8)\j)) );
vector(60, n, a(n+4) ) \\ G. C. Greubel, Dec 10 2019
  • Sage
    def a(n):
    if (n<9): return n-4
    else: return sum(floor((n-4)/floor((n-8)/j)) for j in (1..n-8))
[a(n) for n in (5..60)] # G. C. Greubel, Dec 10 2019

A049792 a(n) = Sum_{k=1..n} floor(n/floor(n/k)).

Original entry on oeis.org

1, 3, 7, 11, 18, 24, 34, 43, 55, 66, 82, 94, 113, 129, 150, 167, 192, 211, 239, 261, 290, 315, 349, 374, 410, 440, 478, 509, 552, 583, 629, 665, 711, 750, 802, 838, 893, 937, 992, 1036, 1097, 1141, 1205, 1255, 1317, 1370, 1440
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049790, A049791, A049793, A049794, A049795, A049796.

Programs

  • GAP
    List([1..60], n-> Sum([1..n], j-> Int(n/Int(n/j)) )); # G. C. Greubel, Dec 10 2019
  • Magma
    [(&+[Floor(n/Floor(n/j)): j in [1..n]]): n in [1..60]]; // G. C. Greubel, Dec 10 2019
  • Maple
    seq( add(floor(n/floor(n/j)), j=1..n), n=1..60); # G. C. Greubel, Dec 10 2019
  • Mathematica
    Table[Total[Table[Quotient[n, Quotient[n, k]], {k, n}]], {n, 47}] (* Ivan Neretin, Jul 29 2015 *)
  • PARI
    a(n) = sum(j=1,n, n\(n\j));
vector(60, n, a(n) ) \\ G. C. Greubel, Dec 10 2019
  • Sage
    [sum(floor(n/floor(n/j)) for j in (1..n)) for n in (1..60)] # G. C. Greubel, Dec 10 2019

Formula

a(n) = A049790(n, n).
a(n) = A222548(n) - Sum_{i=1..n} floor(n/i)*floor(n/(i+1)). - Ridouane Oudra, Jun 22 2020
a(n) ~ (zeta(2) - 1) * n^2. - Vaclav Kotesovec, May 28 2021
Showing 1-6 of 6 results.

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