A202998 Previous integer to m^2/n where m is the next odd prime after n (n excluded).

8, 4, 8, 6, 9, 8, 17, 15, 13, 12, 15, 14, 22, 20, 19, 18, 21, 20, 27, 26, 25, 24, 36, 35, 33, 32, 31, 30, 33, 32, 44, 42, 41, 40, 39, 38, 45, 44, 43, 42, 45, 44, 51, 50, 49, 48, 59, 58, 57, 56, 55, 54, 65, 64, 63, 62, 61, 60, 63, 62, 73, 72

Offset: 1

Author

Michael B. Rees and Frank M Jackson, Jan 07 2012

Keywords

• nonn

Comments

It is conjectured by Michael B Rees (Dec 2011) that for any n > 0, A110835(n) >= a(n)>=n. The Sierpinski conjecture states that: "For any n >= 2 and any k such that 1 < k <= n there exists a prime number between (k-1)*n and k*n (inclusively)". Rees has conjectured that: "For any n >= 1 and any k such that 1 < k < m^2/n where m is the next odd prime after n (n excluded), there exists a prime number between (k-1)*n and k*n (inclusively)".

Examples

			For n=5, a(5)=9. Thus there is at least one prime between any two terms (5 excluded) in the arithmetic progression 5,10,....,45. Note that the progression continues to 5*A110835(5)=90 before there is no prime between 90 and 95. So A110835(5)=18 and 18>=9>=5.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A110835

Programs

Mathematica

nextprime[n_] := (If[n==1, 3, (j=n+1; While[!PrimeQ[j], j++]; j)]);Table[If[IntegerQ[nextprime[i]^2/i], nextprime[i]^2/i-1, Floor[nextprime[i]^2/i]], {i, 1, 100}]
Join[{8},Table[Floor[NextPrime[n]^2/n],{n,2,70}]] (* Harvey P. Dale, Apr 27 2015 *)

Formula

a(n) = previousinteger(nextprime(n)^2/n) where the nextprime(n) excludes n and where previousinteger(i) gives i-1 when i is an integer.