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

%I A202998 #33 Feb 17 2025 02:55:45
%S A202998 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,
%T A202998 32,31,30,33,32,44,42,41,40,39,38,45,44,43,42,45,44,51,50,49,48,59,58,
%U A202998 57,56,55,54,65,64,63,62,61,60,63,62,73,72
%N A202998 Previous integer to m^2/n where m is the next odd prime after n (n excluded).
%C A202998 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)".
%H A202998 Harvey P. Dale, <a href="/A202998/b202998.txt">Table of n, a(n) for n = 1..1000</a>
%F A202998 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.
%e A202998 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.
%t A202998 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}]
%t A202998 Join[{8},Table[Floor[NextPrime[n]^2/n],{n,2,70}]] (* _Harvey P. Dale_, Apr 27 2015 *)
%Y A202998 Cf. A110835
%K A202998 nonn
%O A202998 1,1
%A A202998 Michael B. Rees and _Frank M Jackson_, Jan 07 2012