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).

13, 13, 29, 13, 193, 97, 97, 277, 457, 1193, 109, 229, 937, 397, 349, 1597, 2137, 937, 5569, 5737, 2833, 1549, 6733, 7477, 5077, 3457, 877, 4153, 12277, 11113, 8689, 14029, 11113, 5233, 24109, 14737, 26713, 1297, 77797, 12097, 51577, 57973, 33409, 30493, 49429, 112237, 10333, 143137

Offset

2,1

Comments

The limit of a(n) as n goes to infinity is infinity.

Conjecture: for n >= 12, every a(n) is the lesser of a pair of cousin primes p and p+4, (see A023200).

Links

Table of n, a(n) for n=2..49.

Mathematica

myPrime=Select[Table[Prime[n], {n, 3000000}], Mod[#, 4]==1&];
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)];
myPrimeQ[n_]:=binarySearch[myPrime, n];
nextMyPrime[n_, offset_Integer:1]:=myPrime[[myPrimeQ[NextPrime[n, NestWhile[#1+1&, 1, !myPrimeQ[NextPrime[n, #1]]>0&]]]+offset-1]];
z=1; (*contains exactly ONE myPrime in the interval*)
Table[myPrime[[NestWhile[#1+1&, 1, !((nextMyPrime[n myPrime[[#1]], z+1]>n myPrime[[#1+1]]))&]]], {n, 2, 30}]

Crossrefs

Cf. A195325, A207820, A210465, A210467.

Sequence in context: A166545 A022347 A026907 * A304268 A214466 A302678

Adjacent sequences: A210472 A210473 A210474 * A210476 A210477 A210478

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 23 2013

Status

approved