A268083 Numbers k that are not prime powers and such that gcd(binomial(2*k-1,k), k) = 1.

39, 55, 93, 111, 119, 155, 161, 185, 253, 275, 279, 305, 327, 333, 351, 363, 377, 403, 407, 413, 497, 511, 517, 533, 559, 629, 635, 649, 655, 685, 689, 697, 707, 741, 749, 755, 779, 785, 791, 813, 817, 849, 871, 893, 901, 905, 923, 981, 1003, 1011, 1027, 1043

Offset: 1

Author

Michel Marcus, Jan 26 2016

Keywords

• nonn

Comments

It seems there is a typo in the Gua and Zeng link, it gives 175 instead of 185 as a term.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Victor J.W. Guo and Jiang Zeng, Factors of binomial sums from the Catalan triangle, Journal of Number Theory 130 (2010) 172-186.

Crossrefs

Cf. A000961, A088218, A268082.

Programs

Magma

[n : n in [2..2000] | not IsPrimePower(n) and Gcd(Binomial(2*n-1,n), n) eq 1]; // Vincenzo Librandi, Jan 26 2016

Mathematica

Select[Range[2,1100],!PrimePowerQ[#]&&GCD[Binomial[2#-1,#],#]==1&] (* Harvey P. Dale, May 26 2020 *)

PARI

isok(n) = (n != 1) && !isprimepower(n) && (gcd(binomial(2*n-1,n), n) == 1);

Python

from math import gcd
from sympy import factorint
A268083_list, b = [], 1
for n in range(1,10**4):
    if len(factorint(n)) > 1 and gcd(b,n) == 1:
        A268083_list.append(n)
    b = b*2*(2*n+1)//(n+1) # Chai Wah Wu, Jan 26 2016