Christopher Hunt Gribble - cp's OEIS frontend

A262598 Small positive integer solutions of the simultaneous equations y = ax + b and y^2 = ax^3 + b.

1, 2, 1, 3, 1, 6, 9, 15, 2, 5, 6, 16, 2, 6, 9, 21, 2, 10, 25, 45, 2, 57, 495, 609, 2, 637, 21463, 22737, 3, 10, 25, 55, 3, 14, 49, 91, 4, 14, 49, 105, 4, 18, 81, 153, 5, 13, 40, 105, 5, 18, 81, 171, 5, 22, 121, 231, 5, 574, 27881, 30751, 6, 22, 121, 253, 6, 26, 169, 325

Offset: 1

Adam Kertesz and Christopher Hunt Gribble, Sep 25 2015

The b-file contains all solutions for 1<=a<=100 and 1<=x<=2000000, double-checked by APL and Maple programs.
Of the 224 solutions:
100 are of the form (a, 2(2a-1), (2a-1)^2, (4a-1)(2a-1));
100 are of the form (a, 2(2a+1), (2a+1)^2, (4a+1)(2a+1));
6 are of the form (F(k), F(k+2), F(k)*F(k+1), F(k)*F(k+3)),
where F(n) is the n-th Fibonacci number and k is odd;
(1, 2, 1, 3) is of both forms (a, 2(2a-1), (2a-1)^2, (4a-1)(2a-1)) and (F(k), F(k+2), F(k)*F(k+1), F(k)*F(k+3));
19 have the form (a, cs, c^2t, cu) but appear to be otherwise unrelated.

			The sequence is the result of concatenating the rows in the table of solutions for a, x, b and y. The table starts:
.  a      x      b      y
.  1      2      1      3
.  1      6      9     15
.  2      5      6     16
.  2      6      9     21
.  2     10     25     45
.  2     57    495    609
.  2    637  21463  22737
.  3     10     25     55
.  3     14     49     91
.  4     14     49    105
.  4     18     81    153
.  5     13     40    105
.  5     18     81    171
.  5     22    121    231
.  5    574  27881  30751
.  6     22    121    253
.  6     26    169    325
For a=7 the three solution quadruplets are (7,26,169,351), (7,30,225,435) and (7,1340,120400,129780).

Christopher Hunt Gribble, Table of n, a(n) for n = 1..896

for a to 100 do
   for x to 2000000 do
      y := ceil(sqrt(a*x^3));
      b := y^2-a*x^3;
      if b > 0 and y = a*x+b then
         printf("%10d   %10d   %10d   %10d\n", a, x, b, y);
      end if;
   end do;
end do;

A259301 Taken over all those prime-partitionable numbers m for which there exists a 2-partition of the set of primes < m that has one subset containing two primes only, a(n) is the frequency with which the smaller prime occurs, where n is the prime index.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 3, 2, 4, 4, 3, 4, 5, 7, 8, 5, 8, 7, 8, 9, 10, 10, 11, 12, 12, 14, 13, 13, 12, 15, 14, 14, 17, 14, 19, 17, 12, 18, 13, 19, 20, 22, 20, 23, 21, 15, 21, 21, 23, 25, 26, 23, 26, 26, 19, 23, 27, 24, 29, 27, 26, 28, 31, 29, 30, 25, 30, 29, 34, 30

Offset: 1

Christopher Hunt Gribble, Jun 23 2015

nonn

A number n is called a prime partitionable number if there is a partition {P1,P2} of the primes less than n such that for any composition n1+n2=n, either there is a prime p in P1 such that p | n1 or there is a prime p in P2 such that p | n2.
To demonstrate that a positive integer m is prime-partitionable, a suitable 2-partition {P1, P2} of the set of primes < m must be found. In this sequence we are interested in prime-partitionable numbers such that P1 contains 2 odd primes.
Conjecture: If P1 = {p1a, p1b} with p1a and p1b odd primes, p1a < p1b and p1b = 2*k*p1a + 1 for some positive integer k such that 2*k <= p1a - 3 and if m = p1a + p1b then m is prime-partitionable.

			The table below shows all p1a and p1b pairs for p1a <= 29 that demonstrate that m is prime-partitionable.
. n    p1a    p1b     2k      m
. 3      5     11      2     16
. 4      7     29      4     36
. 5     11     23      2     34
.       11     67      6     78
.       11     89      8    100
. 6     13     53      4     66
.       13     79      6     92
.       13    131     10    144
. 7     17    103      6    120
.       17    137      8    154
.       17    239     14    256
. 8     19    191     10    210
.       19    229     12    248
. 9     23     47      2     70
.       23    139      6    162
.       23    277     12    300
.       23    461     20    484
.10     29     59      2     88
.       29    233      8    262
.       29    349     12    378
.       29    523     18    552
By examining the p1a column it can be seen that
a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 3, a(6) = 3,
a(7) = 3, a(8) = 2, a(9) = 4, a(10) = 4.

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9592

Cf. A059756, A245664.

# Makes use of conjecture in COMMENTS section.
ppgen := proc (ub)
  local freq_p1a, i, j, k, nprimes, p1a, p1b, pless;
  # Construct set of primes < ub in pless.
  pless := {};
  for i from 3 to ub do
    if isprime(i) then
      pless := `union`(pless, {i});
    end if
  end do;
  nprimes := numelems(pless);
  # Determine frequency of each p1a.
  printf("0, ");    # For prime 2.
  for j to nprimes do
    p1a := pless[j];
    freq_p1a := 0;
    for k to (p1a-3)/2 do
      p1b := 2*k*p1a+1;
      if isprime(p1b) then
        freq_p1a := freq_p1a+1;
      end if;
    end do;
    printf("%d, ", freq_p1a);
  end do;
end proc:
ub := 1000:
ppgen(ub):

A259560 Primes p such that p = 2kq + 1 for k a positive integer, q an odd prime and 2k <= q - 3.

Original entry on oeis.org

11, 23, 29, 47, 53, 59, 67, 79, 83, 89, 103, 107, 131, 137, 139, 149, 167, 173, 179, 191, 223, 227, 229, 233, 239, 263, 269, 277, 283, 293, 311, 317, 347, 349, 359, 367, 373, 383, 389, 431, 439, 461, 467, 479, 499, 503, 509, 523, 557, 563, 569, 587, 593, 607

Offset: 1

Christopher Hunt Gribble, Jun 30 2015

nonn

This sequence is associated with the conjecture in A245664 that p + q is prime-partitionable.

There are 138438 values of p in the first 216816 primes, i.e., 63.85%, all of which are distinct.

			The table lists values of n, q, 2k and p for 1 <= n <= 20.
.n      q     2k      p (a(n))
.1      5      2     11
.2     11      2     23
.3      7      4     29
.4     23      2     47
.5     13      4     53
.6     29      2     59
.7     11      6     67
.8     13      6     79
.9     41      2     83
10     11      8     89
11     17      6    103
12     53      2    107
13     13     10    131
14     17      8    137
15     23      6    139
16     37      4    149
17     83      2    167
18     43      4    173
19     89      2    179
20     19     10    191

Christopher Hunt Gribble, Table of n, a(n) for n = 1..20000

Cf. A059756, A245664.

ppgen := proc (n)
  local i, j, k, nprimes, p1a, p1b, p1b_ind, pless, pless_idx, p1b_ind_num_0, p1b_ind_num_1;
  pless := {};
  for i from 3 to n do
    if isprime(i) then
      pless := `union`(pless, {i})
    end if
  end do;
  nprimes := numelems(pless);
  p1b_ind := Vector(nprimes);
  for j to nprimes do
    p1a := pless[j];
    if (1/2)*pless[-1]+1/2 < p1a then
      break
    end if;
    for k to (1/2)*p1a-3/2 do
      p1b := 2*k*p1a+1;
      if member(p1b, pless, 'pless_idx') then
        p1b_ind[pless_idx] := 1
      elif pless[-1] < p1b then
        break
      end if
    end do
  end do;
  p1b_ind_num_0 := 1;
  p1b_ind_num_1 := 0;
  for i to nprimes do
    if p1b_ind[i] = 0 then
      p1b_ind_num_0 := p1b_ind_num_0+1
    else
      p1b_ind_num_1 := p1b_ind_num_1+1;
      fprintf(fop, "%d %d\n", p1b_ind_num_1, pless[i])
    end if
  end do
end proc;
n := 376200;
ppgen(n);

is(n)=my(f=factor(n\2)[,1]); for(i=1,#f, if(n\2/f[i]*2<=f[i]-3, return(isprime(n)))); 0 \\ Charles R Greathouse IV, Jul 15 2015

A257523 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 4, 1, 6, 6, 1, 6, 14, 1, 8, 28, 1, 8, 44, 1, 10, 66, 20, 1, 10, 90, 64, 1, 12, 120, 168, 1, 12, 152, 320, 1, 14, 190, 572, 72, 1, 14, 230, 896, 328, 1, 16, 276, 1360, 984, 1, 16, 324, 1920, 2264, 1, 18, 378, 2660, 4528, 272

Offset: 4

Christopher Hunt Gribble, Apr 27 2015

			The first 9 rows of T(n,k) are:
.\ k    0      1      2     3
n
4       1      2
5       1      2
6       1      4
7       1      4
8       1      6      6
9       1      6     14
10      1      8     28
11      1      8     44
12      1     10     66    20
13      1     10     90    64
14      1     12    120   168
15      1     12    152   320

Andrew Howroyd, Table of n, a(n) for n = 4..860
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592

T(n,k)={(4^k*binomial(n-3*k,k) + ((n%2==0||k%2==0)+(k%2==0)+(k==0)) * 4^((k+1)\2)*binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
for(n=4,15,for(k=0,(n\4), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Terms a(24) and beyond by Andrew Howroyd, May 29 2017

A245372 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has a smallest subset containing four primes only.

Original entry on oeis.org

46, 76, 96, 106, 134, 142, 146, 204, 218, 276, 310, 408, 438, 466, 518, 534, 536, 546, 580, 624, 650, 672, 680, 694, 792, 800, 896, 970, 1000, 1016, 1100, 1160, 1170, 1318, 1344, 1358, 1364, 1384, 1470, 1480

Offset: 1

Christopher Hunt Gribble, Nov 12 2014

nonn

Prime-partitionable numbers are defined in A059756.
To demonstrate that a number is prime-partitionable a suitable 2-partition {P1, P2} of the set of primes < a(n) must be found. In this sequence we are interested in prime-partitionable numbers such that the smallest P1 contains 4 odd primes.
Conjecture:
If P1 = {p1a, p1b, p1c, p1d} with p1a, p1b, p1c and p1d odd primes and p1a < p1b < p1c < p1d then the union of the integer solutions to the ten equation groups below, {{m1}, {m2}, {m3}, {m4}, {m5}, {m6}, {m7}, {m8}, {m9}, {m10}}, contains all even members of {a(n)}:
m1 =  v1*p1a+1   =  v2*p1b+p1a   =  v3*p1c+p1b   =  v4*p1d+p1c
m2 =  v5*p1a+1   =  v6*p1b+p1a^2 =  v7*p1c+p1b   =  v8*p1d+p1a
m3 =  v9*p1a+1   = v10*p1b+p1a^3 = v11*p1c+p1a^2 = v12*p1d+p1a
m4 = v13*p1a+1   = v14*p1b+p1c   = v15*p1c+p1a   = v16*p1d+p1b
m5 = v17*p1a+1   = v18*p1b+p1c   = v19*p1c+p1a^2 = v20*p1d+p1a
m6 = v21*p1a+p1b = v22*p1b+1     = v23*p1c+p1a   = v24*p1d+p1c
m7 = v25*p1a+p1b = v26*p1b+1     = v27*p1c+p1a^2 = v28*p1d+p1a
m8 = v29*p1a+p1b = v30*p1b+p1c   = v31*p1c+1     = v32*p1d+p1a
m9 = v33*p1a+p1c = v34*p1b+1     = v35*p1c+p1b   = v36*p1d+p1a
m10 = v37*p1a+p1c = v38*p1b+p1a  = v39*p1c+1     = v40*p1d+p1b
where the vi, i = 1..40 are constrained odd naturals.

			a(1) = 46 because A245602(5) = 46 and the 2-partition {3, 19, 37, 43} {2, 5, 7, 11, 13, 17, 23, 29, 31, 41} of the set of primes < 46 demonstrates it.

Christopher Hunt Gribble, Prime-partitionable numbers with #P1 = 4
W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272.
R. J. Mathar and M. F. Hasler, Is 52 prime-partitionable?, Seqfan thread (Jun 29 2014), arXiv:1510.07997
W. T. Trotter, Jr. and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142 DOI:10.1002/jgt.3190020206.

Cf. A059756, A244640, A245664, A249302.

prime_part(n)=
{
  my (P = primes(primepi(n-1)));
  for (k1 = 2, #P - 1,
    for (k2 = 1, k1 - 1,
      for (k3 = 1, k2 - 1,
        for (k4 = 1, k3 - 1,
          mask = 2^k1 + 2^k2 + 2^k3 + 2^k4;
          P1 = vecextract(P, mask);
          P2 = setminus(P, P1);
          for (n1 = 1, n - 1,
            bittest(n - n1, 0) || next;
            setintersect(P1, factor(n1)[,1]~) && next;
            setintersect(P2, factor(n-n1)[,1]~) && next;
            next(2)
               );
          print1(n, ", ");
        );
      );
    );
  );
}
# PP = {{2x, x = 1:1000} - {A245664(n), n = 1:145}
#                        - {A249302(n), n = 1:77}}
PP = [2, 4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, \
      38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, \
      ...
      1994, 1996, 1998, 2000];
for(m=1,#PP,prime_part(PP[m]));

A249302 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has a smallest subset containing three primes only.

Original entry on oeis.org

22, 130, 222, 246, 280, 286, 288, 320, 324, 326, 356, 416, 426, 454, 470, 494, 516, 528, 556, 590, 612, 634, 670, 690, 738, 746, 804, 818, 836, 838, 870, 900, 902, 904, 922, 936, 1002, 1026, 1074, 1106, 1116, 1140, 1144, 1150, 1206, 1208, 1262, 1264, 1326, 1338

Offset: 1

Christopher Hunt Gribble, Oct 24 2014

nonn

Prime-partitionable numbers are defined in A059756.
To demonstrate that a number is prime-partitionable a suitable 2-partition {P1, P2} of the set of primes < a(n) must be found. In this sequence we are interested in prime-partitionable numbers such that the smallest P1 contains 3 odd primes.
Conjecture:
If P1 = {p1a, p1b, p1c} with p1a, p1b and p1c odd primes and p1a < p1b < p1c then the union of the integer solutions to the three equation groups below, {{m1}, {m2}, {m3}}, contains all even members of {a(n)}:
m1 = v1*p1a + 1   = v2*p1b + p1a = p1c + p1b
m2 = v3*p1a + 1   = p1b + p1a^2  = p1c + p1a
m3 = v4*p1a + p1b = v5*p1b + 1   = p1c + p1a
where v1, v2, v3, v4 and v5 are odd naturals.

			a(1) = 22 because A059756(2) = 22 and both the 2-partitions {3, 13, 19}, {2, 3, 11, 13, 19} and {5, 7, 17}, {2, 5, 7, 11, 17} of the set of primes < 22 demonstrate it.

Christopher Hunt Gribble, Table of n, a(n) for n = 1..77
Christopher Hunt Gribble, Prime-partitionable numbers with min(#P1)=3
W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272.
R. J. Mathar and M. F. Hasler, Is 52 prime-partitionable?, Seqfan thread (Jun 29 2014), arXiv:1510.07997
W. T. Trotter, Jr. and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142 DOI:10.1002/jgt.3190020206.

Cf. A059756, A244640, A245664.

prime_part(n)=
{
  my (P = primes(primepi(n-1)));
  for (k1 = 2, #P - 1,
    for (k2 = 1, k1 - 1,
      for (k3 = 1, k2 - 1,
        mask = 2^k1 + 2^k2 + 2^k3;
        P1 = vecextract(P, mask);
        P2 = setminus(P, P1);
        for (n1 = 1, n - 1,
          bittest(n - n1, 0) || next;
          setintersect(P1, factor(n1)[,1]~) && next;
          setintersect(P2, factor(n-n1)[,1]~) && next;
          next(2)
        );
        print1(n, ", ");
      );
    );
  );
}
# PP = {{2x, x = 1:1000} - {A245664(n), 1:145}}
PP=[2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, \
    32, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, \
    ...
    1980, 1984, 1986, 1988, 1990, 1994, 1996, 1998, 2000];
for(m=1,#PP,prime_part(PP[m]));

A248027 Sum over each antidiagonal of A248017.

Original entry on oeis.org

0, 0, 0, 4, 69, 554, 3100, 13288, 47492, 147050, 407568, 1030912, 2419025, 5324684, 11099416, 22065120, 42085344, 77378556, 137705904, 237996060, 400624581, 658434694, 1058839380, 1669118984, 2583424948, 3931632406, 5890783808, 8699293304, 12674960961

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

nonn

			a(1)..a(9) are formed as follows:
.             Antidiagonals of A248017                 n   a(n)
.                         0                             1      0
.                      0     0                          2      0
.                   0     0     0                       3      0
.                0     2     2     0                    4      4
.             1    14    39    14     1                 5     69
.          3    66   208   208    66     3              6    554
.      12   198   794  1092   794   198    12           7   3100
.   28   508  2196  3912  3912  2196   508    28        8  13288
.66  1092  5231 10626 13462 10626  5231  1092    66     9  47492

Christopher Hunt Gribble, Table of n, a(n) for n = 1..10000

Cf. A248017.

b := proc (n::integer, k::integer)::integer;
  (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n
   + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4
   - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3
   - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2
   + 48*k + 48*n + 45
   + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3
      + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n
      - 45)*(-1)^k
   + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n
      - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n - 48*k
      - 45)*(-1)^n
   + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
end proc;
for j to 10000 do
  a := 0;
  for k from j by -1 to 1 do
    n := j-k+1;
    a := a+b(n, k);
  end do;
printf("%d, ", a);
end do;

Empirically, a(n) = (2*n^11 + 22*n^10 + 22*n^9 - 462*n^8 - 1122*n^7 + 7392*n^6 - 3509*n^5 - 25663*n^4 + 48950*n^3 - 22869*n^2 - 65133*n + 41580 - (693*n^5 + 3465*n^4 - 6930*n^3 - 45045*n^2 + 27027*n + 41580)*(-1)^n)/2661120.

Empirical g.f.: -x^4*(x^11 + 2*x^10 - 7*x^9 - 10*x^8 - 28*x^7 - 170*x^6 - 484*x^5 - 538*x^4 - 461*x^3 - 176*x^2 - 45*x - 4) / ((x - 1)^12*(x + 1)^6). - Colin Barker, Apr 21 2015

Terms corrected and extended by Christopher Hunt Gribble, Apr 17 2015

A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 14, 39, 14, 1, 3, 66, 208, 208, 66, 3, 12, 198, 794, 1092, 794, 198, 12, 28, 508, 2196, 3912, 3912, 2196, 508, 28, 66, 1092, 5231, 10626, 13462, 10626, 5231, 1092, 66, 126, 2156, 10808, 24648, 35787, 35787, 24648, 10808, 2156, 126

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

			T(n,k) for 1<=n<=8 and 1<=k<=8 is:
.  k   1      2      3      4      5      6      7       8 ...
n
1      0      0      0      0      1      3     12      28
2      0      0      2     14     66    198    508    1092
3      0      2     39    208    794   2196   5231   10808
4      0     14    208   1092   3912  10626  24648   50344
5      1     66    794   3912  13462  35787  81648  164980
6      3    198   2196  10626  35787  94248 212988  428076
7     12    508   5231  24648  81648 212988 477903  955856
8     28   1092  10808  50344 164980 428076 955856 1906128

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n
   + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4
   - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3
   - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2
   + 48*k + 48*n + 45
   + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3
      + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n
      - 45)*(-1)^k
   + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n
      - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n
      - 48*k - 45)*(-1)^n
   + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4 - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3 - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2 + 48*k + 48*n + 45
+ (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3 + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n - 45)*(-1)^k
+ (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n - 48*k - 45)*(-1)^n
+ (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
T(1,k) = A005995(k-5) = (k-3)*(k-1)*((k-4)*(k-2)*2*k + 15*(1-(-1)^k))/480;
T(2,k) = A222715(k) = (k-2)*(k-1)*((2*k-3)(2*k-1)*2*k + 15*(1-(-1)^k))/120.

Terms corrected and extended by Christopher Hunt Gribble, Apr 16 2015

A248060 Sums over successive antidiagonals of A248059.

Original entry on oeis.org

0, 0, 1, 14, 89, 416, 1526, 4740, 12898, 31680, 71527, 150722, 299571, 566592, 1026524, 1791528, 3025188, 4961280, 7926621, 12370710, 18901069, 28327904, 41716466, 60451820, 86313734, 121567680, 169068835, 232386570, 315945319, 425191040, 566777976, 748786896

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

nonn

			a(1..9) are formed as follows:
.             Antidiagonals of A248059                   n    a(n)
.                         0                              1       0
.                      0     0                           2       0
.                   0     1     0                        3       1
.                1     6     6     1                     4      14
.             3    22    39    22     3                  5      89
.          9    60   139   139    60     9               6     416
.      19   135   371   476   371   135    19            7    1526
.   38   266   813  1253  1253   813   266    38         8    4740
.66   476  1574  2706  3254  2706  1574   476    66      9   12898

Christopher Hunt Gribble, Table of n, a(n) for n = 1..10000

Cf. A248059.

b := proc (n::integer, k::integer)::integer;
  (4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 +
   12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 +
   40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 +
   24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 +
   24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k +
   (- 6*k*n + 9)*(-1)^k*(-1)^n)/384
end proc;
for j to 40 do a := 0;
  for k from j by -1 to 1 do
    n := j-k+1;
    a := a+b(n, k);
  end do;
  printf("%d, ", a):
end do:

Empirically, a(n) = (2*n^9 + 18*n^8 + 18*n^7 - 210*n^6 + 588*n^5 + 672*n^4 - 3803*n^3 - 1425*n^2 + 3195*n + 945 + 315*n^3*(-1)^n + 945*n^2*(-1)^n - 315*n*(-1)^n - 945*(-1)^n)/120960.

Empirical g.f.: x^3*(x^8-4*x^6+8*x^5+26*x^4+40*x^3+16*x^2+8*x+1) / ((x-1)^10*(x+1)^4). - Colin Barker, Apr 08 2015

Terms corrected and extended by Christopher Hunt Gribble, Apr 06 2015

A248059 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing four 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 6, 6, 1, 3, 22, 39, 22, 3, 9, 60, 139, 139, 60, 9, 19, 135, 371, 476, 371, 135, 19, 38, 266, 813, 1253, 1253, 813, 266, 38, 66, 476, 1574, 2706, 3254, 2706, 1574, 476, 66, 110, 792, 2770, 5199, 6969, 6969, 5199, 2770, 792, 110, 170, 1245

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

			T(n,k) for 1<=n<=9 and 1<=k<=9 is:
   k    1      2      3      4      5      6      7      8       9 ...
n
1       0      0      0      1      3      9     19     38      66
2       0      1      6     22     60    135    266    476     792
3       0      6     39    139    371    813   1574   2770    4554
4       1     22    139    476   1253   2706   5199   9080   14857
5       3     60    371   1253   3254   6969  13294  23102   37637
6       9    135    813   2706   6969  14841  28197  48852   79401
7      19    266   1574   5199  13294  28197  53381  92266  149645
8      38    476   2770   9080  23102  48852  92266 159216  257878
9      66    792   4554  14857  37637  79401 149645 257878  417156

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248060, A248017, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384;
T(1,k) = sum(A005993(i-4),i=1,k)
       = sum((i-2)*(2*(i-3)*(i-1) + 3*(1-(-1)^(i-1)))/24, i=1,k);
T(2,k) = A071239(k-1) = (k-1)*k*((k-1)^2+2)/6.

Terms corrected and extended by Christopher Hunt Gribble, Apr 06 2015

cp's OEIS Frontend

User: Christopher Hunt Gribble

A262598 Small positive integer solutions of the simultaneous equations y = ax + b and y^2 = ax^3 + b.

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A259301 Taken over all those prime-partitionable numbers m for which there exists a 2-partition of the set of primes < m that has one subset containing two primes only, a(n) is the frequency with which the smaller prime occurs, where n is the prime index.

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A259560 Primes p such that p = 2kq + 1 for k a positive integer, q an odd prime and 2k <= q - 3.

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A257523 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

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A245372 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has a smallest subset containing four primes only.

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A249302 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has a smallest subset containing three primes only.

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A248027 Sum over each antidiagonal of A248017.

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A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

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