cp's OEIS Frontend

A210475 Let p_(4,1)(m) be the m-th prime == 1 (mod 4). Then a(n) is the smallest p_(4,1)(m) such that the interval(p_(4,1)(m)*n, p_(4,1)(m+1)*n) contains exactly one prime == 1 (mod 4).

%I A210475 #16 Feb 13 2013 23:58:30
%S A210475 13,13,29,13,193,97,97,277,457,1193,109,229,937,397,349,1597,2137,937,
%T A210475 5569,5737,2833,1549,6733,7477,5077,3457,877,4153,12277,11113,8689,
%U A210475 14029,11113,5233,24109,14737,26713,1297,77797,12097,51577,57973,33409,30493,49429,112237,10333,143137
%N A210475 Let p_(4,1)(m) be the m-th prime == 1 (mod 4). Then a(n) is the smallest p_(4,1)(m) such that the interval(p_(4,1)(m)*n, p_(4,1)(m+1)*n) contains exactly one prime == 1 (mod 4).
%C A210475 The limit of a(n) as n goes to infinity is infinity.
%C A210475 Conjecture: for n >= 12, every a(n) is the lesser of a pair of cousin primes p and p+4, (see A023200).
%t A210475 myPrime=Select[Table[Prime[n],{n,3000000}],Mod[#,4]==1&];
%t A210475 binarySearch[lst_,find_]:=Module[{lo=1,up=Length[lst],v},(While[lo<=up,v=Floor[(lo+up)/2];If[lst[[v]]-find==0,Return[v]];If[lst[[v]]<find,lo=v+1,up=v-1]];0)];
%t A210475 myPrimeQ[n_]:=binarySearch[myPrime,n];
%t A210475 nextMyPrime[n_,offset_Integer:1]:=myPrime[[myPrimeQ[NextPrime[n,NestWhile[#1+1&,1,!myPrimeQ[NextPrime[n,#1]]>0&]]]+offset-1]];
%t A210475 z=1;(*contains exactly ONE myPrime in the interval*)
%t A210475 Table[myPrime[[NestWhile[#1+1&,1,!((nextMyPrime[n myPrime[[#1]],z+1]>n myPrime[[#1+1]]))&]]],{n,2,30}]
%Y A210475 Cf. A195325, A207820, A210465, A210467.
%K A210475 nonn
%O A210475 2,1
%A A210475 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jan 23 2013