A004273 -id:A004273 - cp's OEIS frontend

A252041 Numbers m such that m - 3 divides m^m + 3.

1, 2, 4, 5, 6, 9, 10, 85, 105, 136, 186, 262, 820, 1161, 2626, 2926, 4924, 10396, 11656, 19689, 27637, 33736, 36046, 42886, 42901, 53866, 55189, 82741, 95266, 103762, 106822, 127401, 135460, 251506, 366796, 375220, 413326, 466966, 531445, 553456, 568876

Offset: 1

Juri-Stepan Gerasimov, Dec 12 2014

nonn

Numbers m such that (m^m + 3)/(m - 3) is an integer.
Most but not all terms are congruent to 4 modulo 6. - Robert G. Wilson v, Dec 19 2014
Note that m^m == 3^m (mod m-3). - Robert Israel, Dec 19 2014

			2 is in this sequence because (2^2 + 3)/(2 - 3) = -7 is an integer.
4 is in this sequence because (4^4 + 3)/(4 - 3) = 259 is an integer.
7 is not in the sequence because (7^7 + 3)/4 = 411773/2, which is not an integer.

Robert G. Wilson v, Table of n, a(n) for n = 1..331

Cf. ...............Numbers n such that x divides y, where:
...x......y....k = 0.....k = 1.....k = 2......k = 3.......
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751..this sequence..
..n+k..n^n-k..A000027...A004275...A251603....A251862......
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Dec 28 2014)

[n: n in [4..50000] | Denominator((n^n+3)/(n-3)) eq 1];

select(t -> 3 &^t + 3 mod (t-3) = 0, [1,2,$4..10^6]); # Robert Israel, Dec 19 2014

fQ[n_] := Mod[PowerMod[n, n, n - 3] + 3, n - 3] == 0; Select[Range@ 1000000, fQ] (* Michael De Vlieger, Dec 13 2014; modified by Robert G. Wilson v, Dec 19 2014 *)

isok(n) = (n != 3) && (Mod(n, n-3)^n  == -3); \\ Michel Marcus, Dec 13 2014

More terms from Michel Marcus, Dec 13 2014

A110185 Coefficients of x in the partial quotients of the continued fraction expansion exp(1/x) = [1, x - 1/2, 12x, 5x, 28x, 9x, 44x, 13x, ...]. The partial quotients all have the form a(n)*x except the constant term of 1 and the initial partial quotient which equals (x - 1/2).

Original entry on oeis.org

0, 1, 12, 5, 28, 9, 44, 13, 60, 17, 76, 21, 92, 25, 108, 29, 124, 33, 140, 37, 156, 41, 172, 45, 188, 49, 204, 53, 220, 57, 236, 61, 252, 65, 268, 69, 284, 73, 300, 77, 316, 81, 332, 85, 348, 89, 364, 93, 380, 97, 396, 101, 412, 105, 428, 109, 444, 113, 460, 117, 476

Offset: 0

Paul D. Hanna, Jul 14 2005

Simple continued fraction expansion of 2*(e - 1)/(e + 1) = 2*tanh(1/2) = 1/(1 + 1/(12 + 1/(5 + 1/(28 + ...)))). - Peter Bala, Oct 01 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. continued fraction expansions: A004273 ( tanh(1) ), A204877 ( 3*tanh(1/3) ), A130824 ( tanh(1/2) ).

a(n)=polcoeff(x*(1+12*x+3*x^2+4*x^3)/(1-x^2)^2+x*O(x^n),n)

G.f.: x*((1+3*x^2) + 4*x*(3+x^2))/(1-x^2)^2 = sum_{n>=0} a(n)*x^n.
From Carl R. White, Feb 11 2010: (Start)
a(n) = sign(n) * (2*n+1) * (3*cos(Pi*n)+5)/2.
a(2n+1) = a(2n-1) + 4, a(2n+2) = a(2n) + 16, with a(0)=0, a(1)=1, a(2)=12. (End)
a(n) = (5+3*(-1)^n)*(2*n-1)/2, with a(0)=0. Sum_{i=0..n} a(i) = A085787(A042948(n)). - Bruno Berselli, Jan 20 2012

A204877 Continued fraction expansion of 3*tanh(1/3).

Original entry on oeis.org

0, 1, 27, 5, 63, 9, 99, 13, 135, 17, 171, 21, 207, 25, 243, 29, 279, 33, 315, 37, 351, 41, 387, 45, 423, 49, 459, 53, 495, 57, 531, 61, 567, 65, 603, 69, 639, 73, 675, 77, 711, 81, 747, 85, 783, 89, 819, 93, 855, 97, 891, 101, 927, 105, 963, 109, 999, 113

Offset: 0

Bruno Berselli, Jan 23 2012

The continued fraction expansions of tanh(1) and 2*tanh(1/2) are in A004273 and A110185, respectively.

Bruno Berselli, Table of n, a(n) for n = 0..1000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A004273, A110185.

I:=[0,1,27,5,63]; [n le 5 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..58]];

ContinuedFraction[3 Tanh[1/3], 158]
CoefficientList[Series[x (1 + 27 x + 3 x^2 + 9 x^3) / ((1 - x)^2 (1 + x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 14 2013 *)

makelist(coeff(taylor(x*(1+27*x+3*x^2+9*x^3)/((1-x)^2*(1+x)^2), x, 0, n), x, n), n, 0, 57);

PARI
```
\p232;
       contfrac(3*tanh(1/3))
```

G.f.: x*(1+27*x+3*x^2+9*x^3)/((1-x)^2*(1+x)^2).
E.g.f.: 9-4*exp(-x)*(1+2*x)+5*exp(x)*(-1+2*x).
a(n) = (5+4*(-1)^n)*(2*n-1), with a(0)=0.
a(n) = 2*a(n-2)-a(n-4) for n>4.
a(n) = a(n-2)+A040314(n-2) for n>2.
a(n)*a(n+1) = a(2*n^2).
Sum(a(i), i=0..n) = A195162(A042948(n)).

A251862 Numbers m such that m + 3 divides m^m - 3.

Original entry on oeis.org

3, 7, 10, 27, 727, 1587, 9838, 758206, 789223, 1018846, 1588126, 1595287, 2387206, 4263586, 9494746, 12697378, 17379860, 21480726, 25439767, 38541526, 44219926, 55561536, 62072326, 64335356, 70032586, 83142466, 85409276

Offset: 1

Juri-Stepan Gerasimov, Dec 10 2014

nonn

m such that m+3 divides (-3)^m - 3. - Robert Israel, Dec 14 2014

			3 is in this sequence because 3 + 3 = 6 divides 3^3 - 3 = 24.

Chai Wah Wu, Table of n, a(n) for n = 1..96

Cf. ...............Numbers n such that x divides y, where:
...x.....y......k=0.......k=1.......k=2........k=3........
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751....A252041......
..n+k..n^n-k..A000027...A004275...A251603..this sequence..
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Jan 14 2015)

[n: n in [2..10000] | Denominator((n^n-3)/(n+3)) eq 1];

select(t ->((-3) &^ (t) - 3) mod (t+3) = 0, [$1..10^6]); # Robert Israel, Dec 14 2014

a251862[n_] := Select[Range[n], Mod[PowerMod[#, #, # + 3] - 3, # + 3] == 0 &]; a251862[10^6] (* Michael De Vlieger, Dec 14 2014, after Robert G. Wilson v at A252041 *)

isok(n) = Mod(n, n+3)^n == 3; \\ Michel Marcus, Dec 10 2014

A251862_list = [n for n in range(10**6) if pow(-3, n, n+3) == 3] # Chai Wah Wu, Jan 19 2015

[n for n in range(10^4) if (n + 3).divides((-3)^n - 3)] # Peter Luschny, Jan 17 2015

More terms from Michel Marcus, Dec 10 2014

A290057 Number T(n,k) of X-rays of n X n binary matrices with exactly k ones; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 13, 23, 30, 30, 23, 13, 5, 1, 1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1, 1, 9, 43, 145, 386, 860, 1660, 2838, 4362, 6090, 7779, 9135, 9892, 9892, 9135, 7779, 6090, 4362, 2838, 1660, 860, 386, 145, 43, 9, 1

Offset: 0

Alois P. Heinz, Jul 19 2017

The X-ray of a matrix is defined as the sequence of antidiagonal sums.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 for k>n^2.

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 3,  4,  3,   1;
  1, 5, 13, 23,  30,  30,  23,  13,   5,   1;
  1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1;
  ...

Alois P. Heinz, Rows n = 0..40, flattened
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Index entries for sequences related to binary matrices

Columns k=0-2 give: A000012, A004273, A091823(n-1) for n>1.
Main diagonal gives A290052.
Row sums give A010790.
Cf. A000290, A290058.

b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      add(b(n-j, i-t, 1-t), j=0..min(i, n)))))(i*(i+1-t))
    end:
T:= (n, k)-> b(k, n, 1):
seq(seq(T(n, k), k=0..n^2), n=0..7);

b[n_,i_,t_]:= b[n, i, t] = Function[{m, jm}, If[n>m, 0, If[n==m, 1, Sum[b[n-j, i-t, 1-t], {j, 0, jm}]]]][i*(i+1-t), Min[i, n]]; T[n_, k_]:= b[k, n, 1]; Table[T[n, k], {n, 0, 7}, {k, 0, n^2}] // Flatten (* Jean-François Alcover, Aug 09 2017, translated from Maple *)

T(n,floor(n^2/2)) = A290058(n).

T(n,k) = T(n,n^2-k).

A166264 If the n-th prime is denoted by p(n) then a(j) = frequency with which each distinct value of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) occurs.

Original entry on oeis.org

174195, 6, 16, 25, 31, 34, 41, 37, 68, 45, 47, 85, 68, 95, 93, 83, 73, 101, 103, 85, 115, 109, 106, 154, 107, 132, 159, 114, 163, 179, 128, 132, 216, 164, 120, 209, 150, 119, 237, 216, 175, 228, 150, 221, 222, 192, 214, 262, 241, 185, 289, 196, 181, 379, 189

Offset: 1

Christopher Hunt Gribble, Oct 10 2009

nonn

The table below shows a(j) for each distinct value of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) for 1 <= n <= 348513, with p(348513) = 4999999 (< 5*10^6).
a(1) appears to increase indefinitely, so the static sequence starts at a(2).
   j   (SQN-SQR)/p(n)    a(j)
  --   --------------  ------
        0        174195
        1             6
        3            16
        5            25
        7            31
        9            34
       11            41
       13            37
       15            68
       17            45
       19            47
       21            85
       23            68
       25            95
       27            93
       29            83
       31            73
       33           101
       35           103
       37            85
       39           115
       41           109
       43           106
       45           154
       47           107
       49           132
       51           159
       53           114
       55           163
       57           179
       59           128
       61           132
       63           216
       65           164
       67           120
       69           209
       71           150
       73           119
       75           237
       77           216
       79           175
       81           228
       83           150
       85           221
       87           222
       89           192
       91           214
       93           262
       95           241
       97           185
       99           289
      101           196
      103           181
      105           379
      107           189
      109           209
      111           314
      113           239

Christopher Hunt Gribble, Table of n, a(n) for n = 1..1973.

Cf. A165951, A165974, A004273.

A252606 Numbers j such that j + 2 divides 2^j + 2.

Original entry on oeis.org

3, 4, 16, 196, 2836, 4551, 5956, 25936, 46775, 65536, 82503, 540736, 598816, 797476, 1151536, 3704416, 4290771, 4492203, 4976427, 8095984, 11272276, 13362420, 21235696, 21537831, 21549347, 29640832, 31084096, 42913396, 49960912, 51127259, 55137316, 56786087, 60296571, 70254724, 70836676

Offset: 1

Juri-Stepan Gerasimov, Mar 03 2015

Numbers j such that (2^j + 2)/(j + 2) is an integer. Numbers j such that (2^j - j)/(j + 2) is an integer.
From Robert Israel, Apr 09 2015: (Start)
The even members of this sequence (4, 16, 196, 2836, ...) are the numbers 2*k-2 where k>=3 is odd and 4^k == -8 (mod k).
The odd members of this sequence (3, 4551, 46775, 82503, ...) are the numbers k-2 where k>=3 is odd and 2^k == -8 (mod k). (End)
2^m is in this sequence for m = (2, 4, 16, 36, 120, 256, 456, 1296, 2556, ...), with the subsequence m = 2^k, k = (1, 2, 4, 8, 16, ...). - M. F. Hasler, Apr 09 2015

			3 is in this sequence because (2^3 + 2)/(3 + 2) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..78

Cf. A001477, A004273, A004275, A081765, A213382, A251603.

[n: n in [0..1200000] | Denominator((2^n+2)/(n+2)) eq 1];

select(t -> 2 &^t + 2 mod (t + 2) = 0, [$1..10^6]); # Robert Israel, Apr 09 2015

Select[Range[10^6],IntegerQ[(2^#+2)/(#+2)]&] (* Ivan N. Ianakiev, Apr 17 2015 *)

for(n=1,10^5,if((2^n+2)%(n+2)==0,print1(n,", "))) \\ Derek Orr, Apr 05 2015

is(n)=Mod(2,n+2)^n==-2 \\ M. F. Hasler, Apr 09 2015

A252606_list = [n for n in range(10**4) if pow(2, n, n+2) == n] # Chai Wah Wu, Apr 16 2015

a(17)-a(22) from Tom Edgar, Mar 03 2015

More terms from Chai Wah Wu, Apr 16 2015

A357778 Maximum number of edges in a 5-degenerate graph with n vertices.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235

Offset: 1

Allan Bickle, Oct 13 2022

nonn

A maximal 5-degenerate graph can be constructed from a 5-clique by iteratively adding a new 5-leaf (vertex of degree 5) adjacent to five existing vertices.

This is also the number of edges in a 5-tree with n>5 vertices. (In a 5-tree, the neighbors of a newly added vertex must form a clique.)

			For n < 7, the only maximal 5-degenerate graph is complete.

Allan Bickle, Fundamentals of Graph Theory, AMS (2020).
J. Mitchem, Maximal k-degenerate graphs, Util. Math. 11 (1977), 101-106.

Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.

Number of edges in a maximal k-degenerate graph for k=2..6: A004273, A296515, A113127, A357778, A357779.

a(n) = C(n,2) for n < 7.

a(n) = 5*n-15 for n > 4.

A357779 Maximum number of edges in a 6-degenerate graph with n vertices.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279

Offset: 1

Allan Bickle, Oct 13 2022

nonn

A maximal 6-degenerate graph can be constructed from a 6-clique by iteratively adding a new 6-leaf (vertex of degree 6) adjacent to six existing vertices.

This is also the number of edges in a 6-tree with n>6 vertices. (In a 6-tree, the neighbors of a newly added vertex must form a clique.)

			For n < 8, the only maximal 6-degenerate graph is complete.

Allan Bickle, Fundamentals of Graph Theory, AMS (2020).
J. Mitchem, Maximal k-degenerate graphs, Util. Math. 11 (1977), 101-106.

Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.

Number of edges in a maximal k-degenerate graph for k=2..6: A004273, A296515, A113127, A357778, A357779.

a(n) = C(n,2) for n < 8.

a(n) = 6*n-21 for n > 5.

A166131 a(j) = minimum value of n for each distinct increasing value of (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic residues of prime(n)) / prime(n) for each j.

Original entry on oeis.org

1, 4, 9, 15, 20, 46, 39, 43, 52, 76, 64, 83, 118, 92, 166, 154, 128, 146, 173, 236, 228, 190, 283, 215, 434, 240, 246, 395, 607, 377, 357, 536, 349, 492, 519, 444, 722, 430, 635, 814, 598, 512, 541, 562, 700, 821, 633, 708, 893, 729, 738

Offset: 1

Christopher Hunt Gribble, Oct 07 2009

nonn

			The table below shows for each value of a(j) the corresponding values of prime(a(j)) and (Sum of the quadratic non-residues of prime(a(j)) - Sum of the quadratic residues of prime(a(j))) / prime(a(j))
.
   j      a(j)    prime(a(j))   (SQN-SQR)/prime(a(j))
  --      ----    -----------   ---------------------
   1         1          2          0
   2         4          7          1
   3         9         23          3
   4        15         47          5
   5        20         71          7
   6        46        199          9
   7        39        167         11
   8        43        191         13
   9        52        239         15
  10        76        383         17
  11        64        311         19
  12        83        431         21
  13       118        647         23
  14        92        479         25
  15       166        983         27
  16       154        887         29
  17       128        719         31
  18       146        839         33
  19       173       1031         35
  20       236       1487         37
  21       228       1439         39
  22       190       1151         41
  23       283       1847         43
  24       215       1319         45
  25       434       3023         47
  26       240       1511         49
  27       246       1559         51
  28       395       2711         53
  29       607       4463         55
  30       377       2591         57
  31       357       2399         59
  32       536       3863         61
  33       349       2351         63
  34       492       3527         65
  35       519       3719         67
  36       444       3119         69
  37       722       5471         71
  38       430       2999         73
  39       635       4703         75
  40       814       6263         77
  41       598       4391         79
  42       512       3671         81
  43       541       3911         83
  44       562       4079         85
  45       700       5279         87
  46       821       6311         89
  47       633       4679         91
  48       708       5351         93
  49       893       6959         95
  50       729       5519         97
  51       738       5591         99

Christopher Hunt Gribble, Table of n, a(n) for n = 1..1973.

Cf. A165951, A165974, A004273.

Sequence corrected and comments added by Christopher Hunt Gribble, Oct 10 2009

cp's OEIS Frontend

A252041 Numbers m such that m - 3 divides m^m + 3.

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A110185 Coefficients of x in the partial quotients of the continued fraction expansion exp(1/x) = [1, x - 1/2, 12*x, 5*x, 28*x, 9*x, 44*x, 13*x, ...]. The partial quotients all have the form a(n)*x except the constant term of 1 and the initial partial quotient which equals (x - 1/2).

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A204877 Continued fraction expansion of 3*tanh(1/3).

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A251862 Numbers m such that m + 3 divides m^m - 3.

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A290057 Number T(n,k) of X-rays of n X n binary matrices with exactly k ones; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.

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A166264 If the n-th prime is denoted by p(n) then a(j) = frequency with which each distinct value of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) occurs.

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A252606 Numbers j such that j + 2 divides 2^j + 2.

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A110185 Coefficients of x in the partial quotients of the continued fraction expansion exp(1/x) = [1, x - 1/2, 12x, 5x, 28x, 9x, 44x, 13x, ...]. The partial quotients all have the form a(n)*x except the constant term of 1 and the initial partial quotient which equals (x - 1/2).