A003992 -id:A003992 - cp's OEIS frontend

A240554 Square array of the greatest prime factor of n^k + 1, read by antidiagonals.

1, 2, 1, 3, 2, 1, 2, 5, 2, 1, 5, 5, 3, 2, 1, 3, 17, 7, 17, 2, 1, 7, 13, 13, 41, 11, 2, 1, 2, 37, 7, 257, 61, 13, 2, 1, 3, 5, 31, 313, 41, 73, 43, 2, 1, 5, 13, 43, 1297, 521, 241, 547, 257, 2, 1, 11, 41, 19, 1201, 101, 601, 113, 193, 19, 2, 1, 3, 101, 73, 241

Offset: 1

T. D. Noe, Apr 07 2014

T. D. Noe, Rows n = 1..60 of triangle, flattened

Cf. A003992 (n^k), A014442 (k=2), A081256 (k=3), A096172 (k=4).

Cf. A240548-A240553 (k=5 to 10).

Table[FactorInteger[(n-k)^k + 1][[-1,1]], {n, 12}, {k, n}]

A305559 [0, -1, -1] together with A000290.

Original entry on oeis.org

0, -1, -1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

Offset: 0

Paul Curtz, Jun 21 2018

Squares leading to an autosequence of the first kind.
The third sequence of the array
A060576(n+1)=         0,  1, 1,  1,  1,   1,   1,   1,   1,  1, ...
A289207(n)=      0,   0,  0, 1,  2,  3,   4,   5,   6,   7, ...
a(n)=       0,  -1,  -1,  0, 1,  4,  9,  16,  25,  36, ...
      0,   10,  10,   5,  0, 1,  8, 27,  64, 125, ...
0, -113, -113, -68, -23,  0, 1, 16, 81, 256, ... .
The first full vertical is (-1)^n*A033312(n).
From 0, the first two nonzero antidiagonals are 0, -1, 10, -113, 1526, ... = (-1)^n* A176824(n+1).
See OEIS Wiki, Autosequence.
a(n) difference table:
0, -1, -1, 0, 1, 4, 9, 16, 25, ...
-1, 0,  1, 1, 3, 5, 7,  9, 11, ...
1,  1,  0, 2, 2, 2, 2,  2,  2, ...
0, -1,  2, 0, 0, 0, 0,  0,  0, ...

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003992, A033312, A060576, A176824, A289207, A000290, A113679.

Join[{0,-1,-1},Range[0,100]^2] (* Paolo Xausa, Nov 13 2023 *)

From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(1 - 2*x + x^3 - 2*x^4)/(1 - x)^3.
E.g.f.: 9 + 5*x + x^2 - exp(x)*(9 - 5*x + x^2). (End)

A329940 Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.

Original entry on oeis.org

1, 3, 2, 7, 8, 3, 15, 26, 15, 4, 31, 80, 63, 24, 5, 63, 242, 255, 124, 35, 6, 127, 728, 1023, 624, 215, 48, 7, 255, 2186, 4095, 3124, 1295, 342, 63, 8, 511, 6560, 16383, 15624, 7775, 2400, 511, 80, 9, 1023, 19682, 65535, 78124, 46655, 16806, 4095, 728, 99, 10

Offset: 1

Roy S. Freedman, Nov 24 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B. A relation R is right unique if (a, b1) in R and (a,b2) in R implies b1=b2. T(n,k) is the number of right unique relations and T(k,n) is the number of left unique relations: relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.

			T(n,k) begins:
    1,    2,     3,      4,       5,       6,        7,        8, ...
    3,    8,    15,     24,      35,      48,       63,       80, ...
    7,   26,    63,    124,     215,     342,      511,      728, ...
   15,   80,   255,    624,    1295,    2400,     4095,     6560, ...
   31,  242,  1023,   3124,    7775,   16806,    32767,    59048, ...
   63,  728,  4095,  15624,   46655,  117648,   262143,   531440, ...
  127, 2186, 16383,  78124,  279935,  823542,  2097151,  4782968, ...
  255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, ...

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

Cf. A037205 (main diagonal).

Cf. A003992, A004248, A009998, A009999, A051128, A051129, A095884.

T:= (n, k)-> (k+1)^n-1:
seq(seq(T(1+d-k, k), k=1..d), d=1..12);

T[n_, k_] := (k + 1)^n - 1; Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

MuPAD
```
T:=(n,k)->(k+1)^n-1:
```

T(n,k) = (k+1)^n - 1.

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A240554 Square array of the greatest prime factor of n^k + 1, read by antidiagonals.

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A305559 [0, -1, -1] together with A000290.

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A329940 Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.

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