A309290
Numbers k such that binomial(k^2,k) - k^2 is squarefree.
Original entry on oeis.org
0, 2, 5, 7, 11, 17, 19, 23, 29, 31, 33, 35, 41, 43, 47, 59, 61, 65, 67, 71, 73, 77, 79, 83, 89
Offset: 1
Cf.
A308078 (binomial(n^2,n) - n^n is squarefree),
A309289 (binomial(2n,n) - n^2 is prime).
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[0] cat [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^2)]; // Vincenzo Librandi, Jul 31 2019
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Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^2] &] (* Vincenzo Librandi, Jul 31 2019 *)
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is(n)=issquarefree(binomial(n^2,n)-n^2)
for(n=0,oo, is(n) && print1(n,", "))
A308078
Numbers k such that binomial(k^2,k) - k^k is squarefree.
Original entry on oeis.org
2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 97
Offset: 1
Cf.
A309289 (binomial(2k,k) - k^2 is prime),
A309290 (binomial(k^2,k) - k^2 is squarefree).
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[0] cat [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^n)];
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Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^#] &]
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is(n)=issquarefree(binomial(n^2,n)-n^n)
for(n=0,oo, is(n) && print1(n,", "))
A383476
Numbers k such that binomial(2k,k) + k is prime.
Original entry on oeis.org
1, 3, 5, 23, 55, 61, 191, 1933, 3049, 8011, 10589, 58687, 100469
Offset: 1
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[k: k in [1..1000] | IsPrime(Binomial(2*k,k)+k)];
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Select[Range[3500], PrimeQ[Binomial[2*#, #] + #] &] (* Amiram Eldar, Apr 28 2025 *)
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from math import comb
from gmpy2 import is_prime
def ok(n): return is_prime(comb(2*n, n) + n)
print([k for k in range(3050) if ok(k)]) # Michael S. Branicky, Apr 28 2025
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