A014206 -id:A014206 - cp's OEIS frontend

A368083 Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.

0, 3, 4, 7, 8, 11, 12, 16, 19, 20, 23, 24, 27, 28, 31, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 71, 72, 75, 76, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 111, 112, 115, 119, 120, 123, 124, 127, 131, 132, 135, 139, 140, 143

Offset: 1

Amiram Eldar, Dec 11 2023

Dimitrov (2023) proved that this sequence is infinite and gave the formula for its asymptotic density.

			0 is a term since 0^2 + 0 + 1 = 1 and 0^2 + 0 + 2 = 2 are both squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stoyan Dimitrov, Square-free pairs n^2 + n + 1, n^2 + n + 2, HAL preprint, hal-03735444, 2023; ResearchGate link.

Subsequence of A353886.

Cf. A002061, A005117, A007674, A014206, A335962, A353887, A368084.

Select[Range[0, 150], And @@ SquareFreeQ /@ (#^2 + # + {1, 2}) &]

is(k) = {my(m = k^2 + k + 1); issquarefree(m) && issquarefree(m + 1);}

A097286 Rectangular array T by descending antidiagonals: T(n,k) = rank of k-th n in A097285.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 13, 9, 6, 8, 21, 15, 11, 10, 14, 31, 23, 17, 12, 16, 22, 43, 33, 25, 19, 18, 24, 32, 57, 45, 35, 27, 20, 26, 34, 44, 73, 59, 47, 37, 29, 28, 36, 46, 58, 91, 75, 61, 49, 39, 30, 38, 48, 60, 74, 111, 93, 77, 63, 51, 41, 40, 50, 62, 76, 92, 133, 113, 95, 79, 65, 53, 42, 52, 64, 78, 94, 112

Offset: 1

Clark Kimberling, Aug 05 2004

As a sequence, this is a permutation of the natural numbers.

			Corner:
   1    3    7   13    21    31    43    57    73    91   111
   2    5    9   15    23    33    45    59    75    93   113
   4    6   11   17    25    35    47    61    77    95   115
   8   10   12   19    27    37    49    63    79    97   117
  14   16   18   20    29    39    51    65    81    99   119
  22   24   26   28    30    41    53    67    83   101   121
  32   34   36   38    40    42    55    69    85   103   123
  44   46   48   50    52    54    56    71    87   105   125
  58   60   62   64    66    68    70    72    89   107   127
  74   76   78   80    82    84    86    88    90   109   129
  92   94   96   98   100   102   104   106   108   110   131

Cf. A002061 (row 1), A014206 (column 1), A097285.

s = {1, 2}; Do[s = Join[s, Riffle[Range[n - 1], n], {n}], {n, 3, 12}];
Grid[Table[Flatten[Position[s, n]], {n, 1, 12}]]  (* Clark Kimberling, May 09 2025 *)

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A241807 Numerators of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)) as defined in A241269.

Original entry on oeis.org

1, 1, 2, 7, 11, 2, 11, 29, 37, 23, 28, 67, 79, 23, 53, 121, 137, 77, 86, 191, 211, 29, 127, 277, 301, 163, 176, 379, 407, 109, 233, 497, 529, 281, 298, 631, 667, 88, 371, 781, 821, 431, 452, 947, 991, 259, 541, 1129, 1177, 613, 638

Offset: 0

Jean-François Alcover and Paul Curtz, Apr 29 2014

The subsequence 1, 23, 77, 163, 281, 431, 613, 827, ..., with indices congruent to 1 mod 8, is 16n^2+6n+1, that is, A000124(8n+1)/2 or A014206(8n+1)/4. Its second differences are constant: (16n^2+6n+1)'' = 32.

The sequence A014206/A241807 is integral and consists of the 16-periodic sequence (2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2, ...).

			1/3, 1/6, 2/15, 7/60, 11/105, 2/21, 11/126, 29/360, 37/495, 23/330, ...

Cf. A000124, A014206, A241269, A188135, A054552, A185438.

Table[(n^2+n+2)/((n+1)*(n+2)*(n+3)) // Numerator, {n, 0, 50}]

a(n) = A014206(n)/period 16: repeat 2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2 (conjectured).
a(4k)     = 8*k^2 +2*k +1,
a(4k+2)   = 4*k^2 +5*k +2,
a(4k+3)   = 8*k^2 +14*k +7,
a(8k+1)   = 16*k^2 +6*k +1,
a(16k+5)  = 16*k^2 +11*k +2,
a(16k+13) = 32*k^2 + 54*k +23.

A247890 Number of digits in (R_n)^n.

Original entry on oeis.org

1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 464, 508, 554, 602, 652, 704, 758, 814, 872, 932, 994, 1058, 1124, 1192, 1262, 1334, 1408, 1484, 1562, 1642, 1724, 1808, 1895, 1983, 2073, 2165, 2259, 2355, 2453, 2553, 2655, 2759, 2865

Offset: 1

Derek Orr, Sep 25 2014

R_n is the n-th repunit (i.e., R_n = 11...111 with n 1's).
From David A. Corneth, Jun 27 2016: (Start)
The number of digits of m is floor(log(m)/log(10)) + 1 for m > 0.
R_n = (10^n - 1) / 9 = (10 - 10^(1-n))/9 * 10^(n-1). Its number of digits is floor(n * log((10 - 10^(1-n))/9) / log(10)) + n * (n - 1) + 1. [corrected by Jason Yuen, Nov 11 2024] (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002061, A014206, A002275.

Cf. A055642, A245593.

[#Intseq(Floor((10^n-1)/9)^n): n in [1..50]]; // Marius A. Burtea, May 20 2019

Table[IntegerLength[((10^n - 1)/9)^n], {n, 54}] (* or *)
Table[IntegerLength[FromDigits[Table[1, {n}]]^n], {n, 54}] (* Michael De Vlieger, Jun 27 2016 *)

PARI
```
vector(100,n,#Str(((10^n-1)/9)^n))
```

a(n) = logint(((10 - 10^(1-n))/9)^n\1,10)+n^2-n+1 \\ David A. Corneth, Jun 27 2016

def a(n): return len(str(int("1"*n)**n))
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Apr 20 2022

a(n) = n^2 - n + 1 = A002061(n), for 1 <= n <= 21.
a(n) = n^2 - n + 2 = A014206(n-1), for 22 <= n <= 43.
a(n) = A055642(A245593(n)). - Michel Marcus, Apr 20 2022

Incorrect conjectures removed by Georg Fischer, May 19 2019

A369292 Array read by downward antidiagonals: A(n,k) = -A(n-1,k) + (k+1)*A(n-1,k+1) + A(n-1,k+2) with A(0,k) = 1, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 8, 18, 1, 4, 14, 42, 108, 1, 5, 22, 84, 276, 780, 1, 6, 32, 150, 612, 2160, 6600, 1, 7, 44, 246, 1212, 5220, 19560, 63840, 1, 8, 58, 378, 2196, 11280, 50880, 200760, 693840, 1, 9, 74, 552, 3708, 22260, 118560, 556920, 2299920, 8361360

Offset: 0

Mikhail Kurkov, Jan 24 2024

			Array begins:
=====================================================
n\k|    0     1     2      3      4      5      6 ...
---+-------------------------------------------------
0  |    1     1     1      1      1      1      1 ...
1  |    1     2     3      4      5      6      7 ...
2  |    4     8    14     22     32     44     58 ...
3  |   18    42    84    150    246    378    552 ...
4  |  108   276   612   1212   2196   3708   5916 ...
5  |  780  2160  5220  11280  22260  40800  70380 ...
6  | 6600 19560 50880 118560 252120 496920 919200 ...
  ...

Column k=0 is A144085.

Rows n=0..2 are A000012, A000027(n+1), A014206(n+1).

A(m,n=m)={my(r=vectorv(m+1), v=vector(n+2*m+1,k,1)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-2, k, -v[k] + k*v[k+1] + v[k+2]); r[1+i] = v[1..n+1]); Mat(r)}
{ A(6) } \\ Andrew Howroyd, Jan 24 2024

A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 2, 3, 2, 1, 0, 1, 2, 4, 4, 2, 2, 0, 1, 2, 5, 6, 4, 2, 1, 0, 1, 2, 6, 8, 7, 4, 2, 0, 0, 1, 2, 7, 10, 11, 8, 4, 2, 1, 0, 1, 2, 8, 12, 16, 14, 8, 4, 2, 2, 0, 1, 2, 9, 14, 22, 22, 15, 8, 4, 2, 1, 0, 1, 2, 10, 16, 29, 32, 26, 16, 8, 4, 2, 0

Offset: 0

Stefano Spezia, May 19 2024

A(n,k) is also the size of the Hamming ball in {0,1}^n of radius (k-1)/2 if k is odd and of the union of two Hamming balls in {0,1}^n of radius k/2-1 whose centers are of Hamming distance 1 if k is even.

			The array begins:
  1, 1, 2, 1,  0,  1,  2,  1, ...
  0, 1, 2, 2,  2,  2,  2,  2, ...
  0, 1, 2, 3,  4,  4,  4,  4, ...
  0, 1, 2, 4,  6,  7,  8,  8, ...
  0, 1, 2, 5,  8, 11, 14, 15, ...
  0, 1, 2, 6, 10, 16, 22, 26, ...
  0, 1, 2, 7, 12, 22, 32, 42, ...
  0, 1, 2, 8, 14, 29, 44, 64, ...
  ...

Noga Alon, Zhihan Jin, and Benny Sudakov, The Helly number of Hamming balls and related problems, arXiv:2405.10275 [math.CO], 2024. See p. 3.
S. L. Bezrukov, Specification of all maximal subsets of the unit cube with respect to given diameter. Problemy Peredachi Informatsii, pages 106-109, 1987. On ResearchGate.
G. Katona, Intersection theorems for systems of finite sets, Acta Mathematica Academiae Scientiarum Hungaricae 15, 329-337 (1964).
Daniel J. Kleitman, On a combinatorial conjecture of Erdös, Journal of Combinatorial Theory, Series A 1, 209-214, (1966).

Cf. A000007 (k=0), A000012 (k=1), A000124 (k=5), A000125 (k=7), A005843 (k=4), A006261 (k=11), A007395 (k=2), A008859 (k=13), A011782 (main diagonal), A014206, A046127 (k=8), A059173, A059174, A130130 (n=1), A158411 (n=2), A373006 (antidiagonal sums).

A[n_,k_]:=If[OddQ[k],Sum[Binomial[n,i],{i,0,(k-1)/2}], Binomial[n-1,k/2-1]+Sum[Binomial[n,i],{i,0,k/2-1}]]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(n,k) = Sum_{i=0..(k-1)/2} binomial(n,i) if k is odd;
A(n,k) = binomial(n-1,k/2-1) + Sum_{i=0..k/2-1} binomial(n,i) if k is even.
A(n,3) = n+1.
A(n,6) = A014206(n-1).
A(n,9) = A000127(n+1).
A(n,10) = A059173(n) for n > 0.
A(n,12) = A059174(n) for n > 0.
A(0,k) = A007877(k) for k > 0.

A055630 Table T(k,m) = k^2 + m read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 4, 2, 2, 9, 5, 3, 3, 16, 10, 6, 4, 4, 25, 17, 11, 7, 5, 5, 36, 26, 18, 12, 8, 6, 6, 49, 37, 27, 19, 13, 9, 7, 7, 64, 50, 38, 28, 20, 14, 10, 8, 8, 81, 65, 51, 39, 29, 21, 15, 11, 9, 9, 100, 82, 66, 52, 40, 30, 22, 16, 12, 10, 10, 121, 101, 83, 67, 53, 41, 31, 23, 17, 13

Offset: 0

Henry Bottomley, Jun 05 2000

			Table begins:
..0...1...4...9..16..25..36..49..64..81.100.121.144...
..1...2...5..10..17..26..37..50..65..82.101.122.145...
..2...3...6..11..18..27..38..51..66..83.102.123.146...
..3...4...7..12..19..28..39..52..67..84.103.124.147...
..4...5...8..13..20..29..40..53..68..85.104.125.148...
..5...6...9..14..21..30..41..54..69..86.105.126.149...
..6...7..10..15..22..31..42..55..70..87.106.127.150...
..7...8..11..16..23..32..43..56..71..88.107.128.151...
..8...9..12..17..24..33..44..57..72..89.108.129.152...
..9..10..13..18..25..34..45..58..73..90.109.130.153...
.10..11..14..19..26..35..46..59..74..91.110.131.154...
... - _Philippe Deléham_, Mar 31 2013

First column is A001477, second column is A000027, first row is A000290, second row is A002522, third row (apart from first term) is A010000, main diagonal is A002378, other diagonals include A028387, A028552, A014209, A002061, A014206, A027688-A027694, each row of A055096 (as upper right triangle) is right hand part of some row of this table

Cf. A185910

A144398 Coefficients of a symmetrical polynomial set:( Pascal's triangle with central zeros) p(x,n)=If[n <= 4, Sum[Binomial[n, i]x^i, {i, 0, n}], x^n + nx^(n - 1) + Binomial[n, 2]x^(n - 2) + nx + Binomial[n, 2]*x^2 + 1].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 0, 15, 6, 1, 1, 7, 21, 0, 0, 21, 7, 1, 1, 8, 28, 0, 0, 0, 28, 8, 1, 1, 9, 36, 0, 0, 0, 0, 36, 9, 1, 1, 10, 45, 0, 0, 0, 0, 0, 45, 10, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

Row sums are: (related to A014206)

{1, 2, 4, 8, 16, 32, 44, 58, 74, 92, 112}

			{1},
{1, 1},
{1, 2, 1},
{1, 3, 3, 1},
{1, 4, 6, 4, 1},
{1, 5, 10, 10, 5, 1},
{1, 6, 15, 0, 15, 6, 1},
{1, 7, 21, 0, 0, 21, 7, 1},
{1, 8, 28, 0, 0, 0, 28, 8, 1},
{1, 9, 36, 0, 0, 0, 0, 36, 9, 1},
{1, 10, 45, 0, 0, 0, 0, 0, 45, 10, 1}

Clear[p, n]; p[x_, n_] = If[n <= 4, Sum[Binomial[n, i]*x^i, {i, 0, n}], x^n + n*x^(n - 1) + Binomial[n, 2]*x^(n - 2) + n*x + Binomial[n, 2]*x^2 + 1]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

p(x,n)=If[n <= 4, Sum[Binomial[n, i]*x^i, {i, 0, n}], x^n + n*x^(n - 1) + Binomial[n, 2]*x^(n - 2) + n*x + Binomial[n, 2]*x^2 + 1]; t(n,m)=coefficients(p(x,n)).

A308981 Nonnegative integers k such that k^3 - 2*k^2 + k - 1 is not composite.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 12, 13, 15, 20, 23, 26, 27, 28, 30, 33, 35, 37, 38, 41, 45, 48, 50, 56, 61, 63, 65, 66, 70, 71, 72, 82, 83, 85, 90, 96, 98, 107, 108, 115, 120, 122, 126, 128, 133, 140, 141, 142, 145, 148, 156, 160, 162, 166, 173, 175, 180, 185, 191, 202, 205, 208, 213, 217, 220

Offset: 1

M. F. Hasler, Jul 04 2019

nonn

Apart the three initial terms which lead to +/-1, all other terms lead to prime P(k) = k^3 - 2*k^2 + k - 1.

The polynomial Q = (((x^2-k)^2-k)^2-x-k)/(x^2 - x - k) of degree 6 has two factors of degree <= 3 when k is in A014206. This can only happen when the constant term of Q, which equals -P(k), is not prime. Therefore, A014206 is a subsequence of the complement of this sequence.

R. Israel, in reply to T. Baruchel, A014206 and computer algebra systems, SeqFan list, July 4, 2019.

Cf. A014206.

[0,1,2] cat  [n: n in [0..220] | IsPrime((n^2*(n-2)+n-1))]; // Vincenzo Librandi, Jul 19 2019

Join[{0, 1, 2}, Select[Range[230], PrimeQ[((#^2 (# - 2) + # - 1))] &]] (* Vincenzo Librandi, Jul 19 2019 *)

select( is(k)={k<3||isprime(k^2*(k-2)+k-1)}, [0..200])

cp's OEIS Frontend

A368083 Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.

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A097286 Rectangular array T by descending antidiagonals: T(n,k) = rank of k-th n in A097285.

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A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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A241807 Numerators of c(n) = (n^2+n+2)/((n+1)*(n+2)*(n+3)) as defined in A241269.

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A247890 Number of digits in (R_n)^n.

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A369292 Array read by downward antidiagonals: A(n,k) = -A(n-1,k) + (k+1)*A(n-1,k+1) + A(n-1,k+2) with A(0,k) = 1, n >= 0, k >= 0.

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A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

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A055630 Table T(k,m) = k^2 + m read by antidiagonals.

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A241807 Numerators of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)) as defined in A241269.

A144398 Coefficients of a symmetrical polynomial set:( Pascal's triangle with central zeros) p(x,n)=If[n <= 4, Sum[Binomial[n, i]x^i, {i, 0, n}], x^n + nx^(n - 1) + Binomial[n, 2]x^(n - 2) + nx + Binomial[n, 2]*x^2 + 1].