OFFSET
1,1
COMMENTS
When m and k are both positive integers and k | m, with m/k = n, then T(n,k) = S(m,k) = (m+k) + (m-k) + (m*k) + (m/k) = S(n*k,k) = n*(k+1)^2, problem proposed by Yakov Perelman.
All terms are nonsquarefree (A013929).
REFERENCES
I. Perelman, L'Algèbre Récréative, Chapitre IV, Les équations de Diophante, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
LINKS
Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
Wikipedia, Yakov Perelman.
FORMULA
T(n,k) = n*(k+1)^2.
T(n,n) = (n+1)^3 - (n+1)^2 = A045991(n+1) for n >= 1.
G.f.: x*(1 + y)/((1 - x)^2*(1 - y)^3). - Stefano Spezia, Mar 31 2022
EXAMPLE
Table begins:
n \ k | 1 2 3 4 5 6 7 8 9 10
----------------------------------------------------------------------------
1 | 4 9 16 25 36 49 64 81 100 121
2 | 8 18 32 50 72 98 128 162 200 242
3 | 12 27 48 75 108 147 192 243 300 363
4 | 16 36 64 100 144 196 256 324 400 484
5 | 20 45 80 125 180 245 320 405 500 605
6 | 24 54 96 150 216 294 384 486 600 726
7 | 28 63 112 175 252 343 448 567 700 847
8 | 32 72 128 200 288 392 512 648 800 968
9 | 36 81 144 225 324 441 576 729 900 1089
10 | 40 90 160 250 360 490 640 810 1000 1210
............................................................................
T(3,4) = 75 = 3*(4+1)^2 corresponds to S(3*4,4) = S(12,4) = (12+4) + (12-4) + (12*4) + 12/4 = 75.
S(10,5) = (10+5) + (10-5) + (10*5) + (10/5) = T(10/5,5) = T(2,5) = 72.
MATHEMATICA
T[n_, k_] := n*(k + 1)^2; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Mar 29 2022 *)
CROSSREFS
Cf. A013929.
Cf. A000290 \ {0,1} (row 1), A001105 \ {0,2} (row 2), A033428 \ {0,3} (row 3), A016742 \ {0,4} (row 4), A033429 \ {0,5} (row 5), A033581 \ {0,6} (row 6).
Cf. A008586 \ {0} (column 1), A008591 \ {0} (column 2), A008598 \ {0} (column 3), A008607 \ {0} (column 4), A044102 \ {0} (column 5).
Cf. A045991 \ {0} (diagonal).
KEYWORD
nonn,tabl
AUTHOR
Bernard Schott, Mar 28 2022
STATUS
approved