How the heck did RWG come across it? WFL On 7/19/13, Bill Gosper <billgosper@gmail.com> wrote:
What the heck use is A006005, "The odd prime numbers together with 1"? Claim: the presence of 1 and the absence of 2 suggest these are actually "Numbers 2n+1 such that 0=Mod[Binomial[2n,n]-(-1)^n,2n+1]" See Concrete Math (G,K&P) Exercise 5.110 I.e., A065091 ∪ A163209 ∪ {1}, suggesting that A163209 should really be *1,5907*, *1194649*, 12327121, "Catalan pseudoprimes: (odd) nonprime integers n=2*m+1 satisfying A000108<http://oeis.org/A000108>(m) == (-1)^m * 2 (mod n). " A176997, " Odd numbers n such that (2^(n-1) mod n) = (4^(n-1) mod n) = (8^(n-1) mod n)= .. = (k^(n-1) mod n) for k=2, 4, 8, .., smallest power of 2>n" should be extended through 341 to avoid matching A006005. (Is this not equivalent to "Numbers n such that 1 = (2^(n-1) mod n) = (4^(n-1) mod n) = (8^(n-1) mod n)= .. = (k^(n-1) mod n) for k=2, 4, 8, .., smallest power of 2>n "?)
Likewise, 0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33,... A102781 "Number of even numbers less than the n - th prime." is probably less useful than "Numbers n such that 0=Mod[Binomial[2n,n]-(-1)^n,2n+1]" which also include 2953, (1093^2 - 1)/2, and (3511^2 - 1)/2. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun