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.

A049760 a(n) = Sum_{k=1..n} T(n,k), array T as in A049759.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 8, 10, 11, 12, 13, 13, 31, 37, 31, 41, 42, 47, 58, 60, 82, 86, 95, 76, 125, 123, 140, 103, 115, 134, 188, 229, 235, 213, 186, 239, 264, 283, 244, 243, 263, 342, 369, 430, 387, 407, 473, 413, 446, 489, 522, 492, 558, 570, 569, 547, 692
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Row sums of A049759.

Programs

  • GAP
    List([1..60], n-> Sum([1..n], k-> PowerMod(n,2,k)) ); # G. C. Greubel, Dec 14 2019
  • Magma
    [&+[n^2 mod i: i in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Sep 18 2017
  • Maple
    seq( add( `mod`(n^2, k), k = 1..n), n = 1..60); # G. C. Greubel, Dec 14 2019
  • Mathematica
    Table[Sum[Mod[n^2, i], {i, n}], {n, 60}] (* Vincenzo Librandi, Sep 18 2017 *)
Table[Sum[PowerMod[n,2,k],{k,n}],{n,60}] (* Harvey P. Dale, Mar 29 2018 *)
  • PARI
    a(n)=my(N=n^2);sum(k=2,n-1,N%k) \\ Charles R Greathouse IV, Mar 27 2014
  • Sage
    [sum(power_mod(n,2,k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Dec 14 2019

Formula

a(n) = Sum_{i=1..n} (n^2 mod i). - Wesley Ivan Hurt, Sep 15 2017

A049767 Triangular array T, read by rows: T(n,k) = (k^2 mod n) + (n^2 mod k), for k = 1..n and n >= 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  0;
  1,  0;
  1,  2,  0;
  1,  0,  2,  0;
  1,  5,  5,  2,  0;
  1,  4,  3,  4,  2,  0;
  1,  5,  3,  3,  8,  2,  0;
  1,  4,  2,  0,  5,  8,  2,  0;
  1,  5,  0,  8,  8,  3,  8,  2,  0;
  1,  4, 10,  6,  5, 10, 11,  8,  2,  0;
  ...

Links

Crossrefs

Row sums are in A049768.
Cf. A048152, A049759.

Programs

  • GAP
    Flat(List([1..15], n-> List([1..n], k-> PowerMod(k,2,n) + PowerMod(n,2,k) ))); # G. C. Greubel, Dec 13 2019
  • Magma
    [[Modexp(k,2,n) + Modexp(n,2,k): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Dec 13 2019
  • Maple
    seq(seq( `mod`(k^2, n) + `mod`(n^2, k), k = 1..n), n = 1..15); # G. C. Greubel, Dec 13 2019
  • Mathematica
    Table[PowerMod[k,2,n] + PowerMod[n,2,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Dec 13 2019 *)
  • PARI
    T(n,k) = lift(Mod(k,n)^2) + lift(Mod(n,k)^2);
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 13 2019
  • Sage
    [[power_mod(k,2,n) + power_mod(n,2,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 13 2019

Formula

T(n, k) = A048152(n, k) + A049759(n, k). - Michel Marcus, Nov 21 2019

A049761 Triangular array T, read by rows: T(n,k) = n^3 mod k, for k = 1..n and n >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 3, 3, 1, 0, 0, 0, 2, 0, 2, 2, 1, 0, 0, 1, 0, 1, 4, 3, 1, 1, 0, 0, 0, 1, 0, 0, 4, 6, 0, 1, 0, 0, 1, 2, 3, 1, 5, 1, 3, 8, 1, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 8, 1, 0, 0, 1, 1, 1, 2, 1, 6, 5, 1, 7, 8, 1, 0
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  0;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 1, 2, 1, 0;
  0, 0, 0, 0, 1, 0;
  0, 1, 1, 3, 3, 1, 0;
  0, 0, 2, 0, 2, 2, 1, 0;
  0, 1, 0, 1, 4, 3, 1, 1, 0;
  0, 0, 1, 0, 0, 4, 6, 0, 1, 0;
  0, 1, 2, 3, 1, 5, 1, 3, 8, 1, 0;
  ...

Links

Crossrefs

Row sums are in A049762.
Cf. A049759, A049760, A049763, A049764.

Programs

  • GAP
    Flat(List([1..15], n-> List([1..n], k-> PowerMod(n,3,k) ))); # G. C. Greubel, Dec 13 2019
  • Magma
    [[Modexp(n,3,k): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Dec 13 2019
  • Maple
    seq(seq( `mod`(n^3, k), k = 1..n), n = 1..15); # G. C. Greubel, Dec 13 2019
  • Mathematica
    Table[PowerMod[n,3,k], {n,15}, {k, n}]//Flatten (* G. C. Greubel, Dec 13 2019 *)
  • PARI
    T(n,k) = lift(Mod(n,k)^3);
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 13 2019
  • Sage
    [[power_mod(n,3,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 13 2019

A049763 Triangular array T, read by rows: T(n,k) = n^4 mod k, for k = 1..n and n >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 4, 1, 0, 0, 1, 0, 1, 1, 3, 2, 1, 0, 0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 0, 1, 1, 1, 1, 1, 4, 1, 7, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 6, 1, 0, 0, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  0;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 1, 1, 1, 0;
  0, 0, 0, 0, 1, 0;
  0, 1, 1, 1, 1, 1, 0;
  0, 0, 1, 0, 1, 4, 1, 0;
  0, 1, 0, 1, 1, 3, 2, 1, 0;
  0, 0, 1, 0, 0, 4, 4, 0, 1, 0;
  ...

Links

Crossrefs

Row sums are in A049764.
Cf. A049759, A049760, A049761, A049762.

Programs

  • GAP
    Flat(List([1..15], n-> List([1..n], k-> PowerMod(n,4,k) ))); # G. C. Greubel, Dec 13 2019
  • Magma
    [[Modexp(n,4,k): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Dec 13 2019
  • Maple
    seq(seq( `mod`(n^4, k), k = 1..n), n = 1..20); # G. C. Greubel, Dec 13 2019
  • Mathematica
    Flatten[Table[PowerMod[n,4,k],{n,20},{k,n}]] (* Harvey P. Dale, Jan 19 2015 *)
  • PARI
    T(n,k) = lift(Mod(n,k)^4);
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 13 2019
  • Sage
    [[power_mod(n,4,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 13 2019

A049762 a(n) = Sum_{k=1..n} T(n,k), array T as in A049761.

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 9, 7, 11, 12, 25, 18, 34, 34, 41, 32, 76, 44, 87, 64, 93, 85, 122, 75, 80, 160, 144, 132, 172, 103, 218, 232, 220, 245, 251, 210, 299, 330, 344, 315, 413, 275, 456, 392, 383, 472, 502, 479, 449, 553, 557, 626, 646, 632, 628, 618, 771
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Row sums of A049761.
Cf. A049759, A049760, A049763, A049764.

Programs

  • GAP
    List([1..60], n-> Sum([1..n], k-> PowerMod(n,3,k)) ); # G. C. Greubel, Dec 14 2019
  • Magma
    [&+[n^3 mod i: i in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Sep 18 2017
  • Maple
    seq( add( `mod`(n^3, k), k = 1..n), n = 1..60); # G. C. Greubel, Dec 14 2019
  • Mathematica
    Table[Sum[Mod[n^3, i], {i, n}], {n, 60}] (* Vincenzo Librandi, Sep 18 2017 *)
  • PARI
    vector(60, n, sum(k=1,n, lift(Mod(n,k)^3)) ) \\ G. C. Greubel, Dec 14 2019
  • Sage
    [sum(power_mod(n,3,k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Dec 14 2019

Formula

a(n) = Sum_{i=1..n} (n^3 mod i). - Wesley Ivan Hurt, Sep 15 2017

A096459 Triangle read by rows: T(n,k) = n^2 mod prime(k), 1<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 4, 0, 1, 1, 2, 1, 1, 0, 4, 3, 0, 0, 1, 1, 3, 10, 1, 1, 4, 0, 5, 10, 15, 0, 1, 4, 1, 9, 12, 13, 7, 1, 0, 1, 4, 4, 3, 13, 5, 12, 0, 1, 0, 2, 1, 9, 15, 5, 8, 13, 1, 1, 1, 2, 0, 4, 2, 7, 6, 5, 28, 0, 0, 4, 4, 1, 1, 8, 11, 6, 28, 20, 33, 1, 1, 4, 1, 4, 0, 16, 17, 8, 24, 14, 21, 5, 0, 1, 1, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 12 2004

Keywords

Comments

T(n,k)=0 iff k is a prime factor of n:
A001221(n) = number of zeros in n-th row;
T(n,1)=A000035(n);
T(n,2)=A011655(n) for n>1; T(n,3)=A070430(n) for n>2;
T(n,4)=A053879(n) for n>3; T(n,5)=A070434(n) for n>4;
T(n,6)=A070436(n) for n>5; T(n,7)=A054580(n) for n>6;
T(n,8)=A070441(n) for n>7; T(n,9)=A070445(n) for n>8;
T(n,10)=A070451(n) for n>9;
T(n,n)=A069547(n).

Examples

			Triangle begins:
1;
0, 1;
1, 0, 4;
0, 1, 1, 2;
1, 1, 0, 4, 3;
0, 0, 1, 1, 3, 10;
1, 1, 4, 0, 5, 10, 15;
......

Links

Crossrefs

Cf. A000290, A000040, A049759.

Programs

  • Mathematica
    Table[Mod[n^2, Prime[k]], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, May 20 2017 *)
Showing 1-6 of 6 results.

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