A239623 -id:A239623 - cp's OEIS frontend

A239622 Conjecturally, the irregular triangle of numbers k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

0, 1, 2, 4, 3, 6, 7, 9, 10, 11, 12, 21, 22, 28, 29, 30, 31, 36, 37, 54, 55, 57, 58, 110, 171, 784, 786, 5, 8, 15, 16, 17, 20, 35, 42, 45, 50, 51, 52, 53, 56, 59, 60, 77, 80, 133, 134, 135, 136, 156, 157, 158, 159, 160, 161, 170, 210, 211, 212, 400, 401, 402, 651, 652, 785

Offset: 0

T. D. Noe, Mar 27 2014

Row 0 lists the numbers k such that binomial(2k,k) is squarefree. Sequence A110494 lists the first term of each row; A239623 lists the conjectured last term; A239624 lists the conjectured length of each row.

			The irregular triangle begins:
0, 1, 2, 4
3, 6, 7, 9,..., 784, 786
5, 8, 15, 16,..., 652, 785
13, 14, 18, 19,..., 445, 2080
25, 26, 27, 32,..., 783, 902
61, 62, 63, 64,..., 2033, 2034

T. D. Noe, Rows n = 0..40 of irregular triangle, flattened

Cf. A059097 (union of first two rows), A110493, A110494, A239623, A239624.

b = 1; t = Table[b = b*(4 - 2/n); last = 0; Do[If[Mod[b, p^2] == 0, last = p], {p, Prime[Range[PrimePi[Sqrt[2*n]]]]}]; last, {n, 20000}]; t = Join[{0}, t]; Table[Flatten[Position[t, p]] - 1, {p, Join[{0}, Prime[Range[20]]]}]

A239624 Conjecturally, the number of numbers k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

Original entry on oeis.org

4, 23, 38, 50, 51, 54, 65, 70, 107, 127, 127, 165, 155, 150, 239, 287, 280, 179, 336, 314, 230, 453, 423, 600, 612, 419, 246, 454, 455, 892, 1117, 624, 916, 432, 1115, 363, 934, 1061, 763, 1073, 1203, 524, 1523, 559, 1278, 735, 2221, 1987, 929, 475, 1179, 1605

Offset: 0

T. D. Noe, Mar 27 2014

nonn

The 0th term is the largest number k such that binomial(2k,k) is squarefree.

Cf. A110493, A110494, A239622, A239623.

b = 1; t = Table[b = b*(4 - 2/n); last = 0; Do[If[Mod[b, p^2] == 0, last = p], {p, Prime[Range[PrimePi[Sqrt[2*n]]]]}]; last, {n, 20000}]; t = Join[{0}, t]; Table[Length[Position[t, p]], {p, Join[{0}, Prime[Range[20]]]}]

cp's OEIS Frontend

A239622 Conjecturally, the irregular triangle of numbers k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

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