A234248 -id:A234248 - cp's OEIS frontend

A236249 Primes of the form C(2m, m) - prime(m), where C(2m, m) = (2*m)!/(m!)^2.

3, 241, 911, 184727, 30067266499540931, 1454272161238683681127450712107767894181359647011258114106151524833563647084221

Offset: 1

Zhi-Wei Sun, Jan 21 2014

nonn

Though the primes in this sequence are very rare, according to the conjecture in A236256 there should be infinitely many such primes.
See A236248 for a list of known numbers m with C(2*m, m) - prime(m) prime.
See also A236245 for a similar sequence.

			a(1) = 3 since C(2*1, 1) - prime(1) = 0 is not prime, but C(2*2, 2) - prime(2) = 6 - 3 = 3 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..12

Cf. A000040, A000984, A236241, A236242, A236245, A236248, A236256.

t[n_]:=Binomial[2n,n]-Prime[n]
a[n_]:=t[A234248(n)]
Table[a[n],{n,1,6}]

A244504 Number of distinct lines passing through at least two points in a triangular grid of side n.

Original entry on oeis.org

3, 9, 24, 51, 102, 177, 294, 459, 690, 987, 1380, 1875, 2508, 3279, 4212, 5319, 6648, 8199, 10026, 12141, 14580, 17343, 20496, 24051, 28068, 32547, 37542, 43071, 49218, 55983, 63456, 71661, 80658, 90447, 101100, 112635, 125160, 138675, 153252, 168915, 185784

Offset: 2

Heinrich Ludwig, Sep 04 2014

nonn

Heinrich Ludwig, Table of n, a(n) for n = 2..1000

Cf. A234248.

g[i_]:=If[i>0,i*(i+1)/2,0]; Table[3*Sum[EulerPhi[j]*(g[n-j]-g[n-2*j]),{j,1,n-1}],{n,2,50}] (* Vaclav Kotesovec, Sep 04 2014 after Jon E. Schoenfield *)

g(j) = if (j > 0, j*(j+1)/2, 0);
a(n) = 3*sum(j = 1, n-1, eulerphi(j)*(g(n-j)-g(n-2*j))); \\ Michel Marcus, Sep 04 2014

a(n) = 3*sum(j = 1..n-1, euler_phi(j)*(g(n-j)-g(n-2*j))), where g(i) = i*(i+1)/2 if i > 0, otherwise 0, after Jon E. Schoenfield.

A362014 Number of distinct lines passing through exactly two points in a triangular grid of side n.

Original entry on oeis.org

0, 0, 3, 6, 18, 39, 81, 141, 237, 369, 561, 801, 1119, 1521, 2043, 2667, 3429, 4329, 5415, 6675, 8163, 9879, 11877, 14127, 16695, 19593, 22881, 26523, 30591, 35085, 40089, 45591, 51681, 58359, 65715, 73701, 82389, 91791, 102015, 113007, 124875

Offset: 0

Caleb Stanford, Apr 03 2023

nonn

Samuel Dittmer, Hiram Golze, Grant Molnar, and Caleb Stanford, Puzzle and Proof: A Decade of Problems from the Utah Math Olympiad, CRC Press, 2025, p. 34.

Cf. A234248, A244504 (lines which contain 2 or more points), A050534 (total number of pairs of points). Both are upper bounds.

a(n) = A244504(n) - A234248(n). - Andrew Howroyd, Apr 03 2023

cp's OEIS Frontend

A236249 Primes of the form C(2m, m) - prime(m), where C(2m, m) = (2*m)!/(m!)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A244504 Number of distinct lines passing through at least two points in a triangular grid of side n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A362014 Number of distinct lines passing through exactly two points in a triangular grid of side n.

Views

Author

Keywords

References

Crossrefs

Formula

cp's OEIS Frontend

A236249 Primes of the form C(2*m, m) - prime(m), where C(2*m, m) = (2*m)!/(m!)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A244504 Number of distinct lines passing through at least two points in a triangular grid of side n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A362014 Number of distinct lines passing through exactly two points in a triangular grid of side n.

Views

Author

Keywords

References

Crossrefs

Formula

A236249 Primes of the form C(2m, m) - prime(m), where C(2m, m) = (2*m)!/(m!)^2.