A048154 -id:A048154 - cp's OEIS frontend

A048155 a(n)=Sum{T(n,k): k=1,2,...,n}, array T as in A048154.

0, 1, 3, 4, 10, 15, 21, 16, 27, 45, 55, 60, 78, 91, 105, 96, 136, 135, 171, 180, 210, 231, 253, 240, 250, 325, 243, 364, 406, 435, 465, 384, 528, 561, 595, 540, 666, 703, 741, 720, 820, 861, 903, 924, 945, 1035, 1081, 1056, 1029

Offset: 1

Clark Kimberling

nonn

			a(3) = 3 since (1^3 mod 3) + (2^3 mod 3) + (3^3 mod 3) = (1 mod 3) + (8 mod 3) + (27 mod 3) = 1 + 2 + 0 = 3.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A048154.

Table[Sum[Mod[k^3, n], {k, n}], {n, 50}] (* or *)
Table[Sum[k (1 - Floor[k^3/n] + Floor[(k^3 - 1)/n]), {k, n}], {n, 50}] (* Michael De Vlieger, Jun 26 2016 *)

a(n) = Sum_{k=1..n} k*(1-floor(k^3/n)+floor((k^3 -1)/n)). - Anthony Browne, Jun 26 2016

A049769 Triangular array T read by rows: T(n,k) = (k^3 mod n) + (n^3 mod k).

Original entry on oeis.org

0, 1, 0, 1, 3, 0, 1, 0, 4, 0, 1, 4, 4, 5, 0, 1, 2, 3, 4, 6, 0, 1, 2, 7, 4, 9, 7, 0, 1, 0, 5, 0, 7, 2, 8, 0, 1, 9, 0, 2, 12, 3, 2, 9, 0, 1, 8, 8, 4, 5, 10, 9, 2, 10, 0, 1, 9, 7, 12, 5, 12, 3, 9, 11, 11, 0, 1, 8, 3, 4, 8, 0, 13, 8, 9, 12, 12, 0

Offset: 1

Clark Kimberling

			Triangle begins as:
  0;
  1, 0;
  1, 3, 0;
  1, 0, 4, 0;
  1, 4, 4, 5, 0;
  1, 2, 3, 4, 6, 0;
  1, 2, 7, 4, 9, 7, 0;
  1, 0, 5, 0, 7, 2, 8, 0;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A048154, A049761, A049767.

Flat(List([1..15], n-> List([1..n], k-> PowerMod(k,3,n) + PowerMod(n,3,k) ))); # G. C. Greubel, Dec 13 2019

[[Modexp(k,3,n) + Modexp(n,3,k): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Dec 13 2019

seq(seq( `mod`(k^3, n) + `mod`(n^3, k), k = 1..n), n = 1..15); # G. C. Greubel, Dec 13 2019

Table[PowerMod[k,3,n] + PowerMod[n,3,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Dec 13 2019 *)

T(n,k) = lift(Mod(k,n)^3) + lift(Mod(n,k)^3);
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 13 2019

[[power_mod(k,3,n) + power_mod(n,3,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 13 2019

T(n, k) = A048154(n, k) + A049761(n, k). - Michel Marcus, Dec 13 2019

cp's OEIS Frontend

A048155 a(n)=Sum{T(n,k): k=1,2,...,n}, array T as in A048154.

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