A180686 Positive integers k such that the number of intersections of diagonals in the interior of a regular k-gon is prime.

5, 6, 14, 24, 44, 58, 72, 76, 80, 84, 86, 104, 128, 134, 138, 180, 186, 188, 218, 228, 246, 256, 266, 280, 300, 320, 352, 360, 380, 390, 408, 450, 480, 508, 518, 524, 526, 532, 546, 548, 552, 576, 584, 590, 604, 616, 630, 656, 658, 686, 712, 724, 726, 728, 730

Offset: 1

Author

Robert G. Wilson v, Sep 16 2010

Keywords

• nonn

References

• Chris K. Caldwill & G. L. Honaker, Jr., Prime Curios!, The Dictionary of Prime Number Trivia, CreateSpace, Sept. 2009, p. 145.

Links

• Robin Visser, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006561.

Programs

Mathematica

del[m_, n_] := If[ Mod[n, m] == 0, 1, 0]; Int[n_] := If[n < 4, 0, Binomial[n, 4] + del[2, n] (-5n^3 + 45n^2 - 70n + 24)/24 - del[4, n] (3n/2) + del[6, n] (-45n^2 + 262n)/6 + del[12, n]*42n + del[18, n]*60n + del[24, n]*35n - del[30, n]*38n - del[42, n]*82n - del[60, n]*330n - del[84, n]*144n - del[90, n]*96n - del[120, n]*144n - del[210, n]*96n]; Select[ Range@ 759, PrimeQ@ Int@# &]

Sage

def is_A180686(k):
    return Integer(binomial(k,4) + (-5*k^3+45*k^2-70*k+24)*(k%2==0)/24
        - 3*k*(k%4==0)/2 + (-45*k^2+262*k)*(k%6==0)/6 + 42*k*(k%12==0)
        + 60*k*(k%18==0) + 35*k*(k%24==0) - 38*k*(k%30==0)
        - 82*k*(k%42==0) - 330*k*(k%60==0) - 144*k*(k%84==0)
        - 96*k*(k%90==0) - 144*k*(k%120==0) - 96*k*(k%210==0)).is_prime()
print([k for k in range(1, 1000) if is_A180686(k)])  # Robin Visser, Jul 29 2024

Extensions

Name edited by Robin Visser, Jul 29 2024