A267824
Composite numbers n such that binomial(2n-1, n-1) == 1 (mod n^2).
Original entry on oeis.org
283686649, 4514260853041
Offset: 1
a(1) = 16843^2 and a(2) = 2124679^2 are squares of Wolstenholme primes A088164.
- Richard J. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica, 71 (1995), 381-389.
- J. Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018.
Cf.
A000984,
A034602,
A082180,
A088164,
A099905,
A099906,
A099907,
A099908,
A136327,
A177783,
A212557,
A228562,
A242473,
A244214,
A244919,
A246130,
A246132,
A246133,
A246134,
A260209,
A260210,
A263429,
A263882,
A281302.
A282410
a(n) = binomial(2*p-1, p-1) modulo p^5, where p = prime(n).
Original entry on oeis.org
3, 10, 126, 1716, 30614, 2198, 1100513, 713337, 4635628, 4511966, 15729649, 49285370, 10820598, 115444165, 110571496, 84562137, 145202954, 386548644, 208729523, 1232287574, 790871562, 2277840181, 3525066856, 4912928962, 7258488370, 8723558568, 9006255935
Offset: 1
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- C. Aebi and G. Cairns, Wolstenholme again, arXiv:1502.05750 [math.NT], 2015.
- R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), 381-389.
-
f[n_] := Block[{p = Prime@n}, Mod[ Binomial[ 2p -1, p -1], p^5]]; Array[f, 27] (* Robert G. Wilson v, Feb 14 2017 *)
Table[Mod[Binomial[2p-1,p-1],p^5],{p,Prime[Range[30]]}] (* Harvey P. Dale, Jul 07 2022 *)
-
a(n) = my(p=prime(n)); lift(Mod(binomial(2*p-1, p-1), p^5))
-
from sympy import Mod, binomial, prime
def A282410(n): return int(Mod(binomial(2*(p:=prime(n))-1,p-1,evaluate=False),p**5)) # Chai Wah Wu, Apr 24 2025
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