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 A343533 a(n) is the largest value of k such that binomial(2*m-1, m-1) == 1 (mod m^k) for m = 2*n + 1. 0
 2, 3, 3, 1, 3, 3, 0, 3, 3, 0, 3, 1, 0, 3, 3, 0, 0, 3, 0, 3, 3, 0, 3, 1, 0, 3, 0, 0, 3, 3, 0, 0, 3, 0, 3, 3, 0, 0, 3, 0, 3, 0, 0, 3, 0, 0, 0, 3, 0, 3, 3, 0, 3, 3, 0, 3, 0, 0, 0, 1, 0, 1, 3, 0, 3, 0, 0, 3, 3, 0, 0, 0, 0, 3, 3, 0, 0, 3, 0, 0, 3, 0, 3, 1, 0, 3, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS If 2*n + 1 is a prime >= 5, then a(n) >= 3 by Wolstenholme's theorem. If 2*n + 1 is a Wolstenholme prime (A088164), then a(n) >= 4. If 2*n + 1 is a term of A267824, then a(n) >= 2. If 2*n + 1 is the square of an odd prime, the cube of a prime >= 5 or a term of A228562, then a(n) >= 1. LINKS MAPLE a := proc(n) local x, x0, y, k, bound; bound := 1000; x := 2*n + 1; x0 := x; y := binomial(4*n + 1, 2*n); for k from 0 to bound while y mod x = 1 do     x := x * x0 od; if k < bound then k else print("No k below ", bound) fi end: seq(a(n), n = 1..100); # Peter Luschny, Apr 22 2021 PROG (PARI) a(n) = my(x=2*n+1, b=binomial(2*x-1, x-1)); for(k=1, oo, if(Mod(b, x^k)!=1, return(k-1))) CROSSREFS Cf. A005408, A088164, A136327, A228562, A267824. Sequence in context: A109199 A279813 A256909 * A308734 A279004 A172528 Adjacent sequences:  A343530 A343531 A343532 * A343534 A343535 A343536 KEYWORD nonn AUTHOR Felix Fröhlich, Apr 18 2021 STATUS approved

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Last modified September 28 01:27 EDT 2021. Contains 347698 sequences. (Running on oeis4.)