login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) is the smallest product of distinct prime pair p*q such that both 2^n*p+q and p+2^n+q are prime numbers.
1

%I #9 May 21 2013 11:58:15

%S 15,15,15,15,15,55,15,15,39,133,35,177,51,57,39,259,65,51,329,39,141,

%T 51,1199,85,341,133,141,415,471,21,515,15,635,247,365,57,501,133,305,

%U 1501,159,111,411,1135,291,505,51,913,515,411,471,849,1569,895,155,1897

%N a(n) is the smallest product of distinct prime pair p*q such that both 2^n*p+q and p+2^n+q are prime numbers.

%H Lei Zhou, <a href="/A225917/b225917.txt">Table of n, a(n) for n = 1..886</a>

%e 15=3*5, both 2*3+5=11 and 3+2*5=13 are prime numbers, so a(1)=15;

%e ...

%e when n=6, for any numbers in the form of p*q that are smaller than 55, 2^6*p+q and p+2^6*q are not both prime numbers. 55=5*11, and 2^6*5+11=331 and 5+2^6*11=709 are prime numbers, so a(6)=55.

%t NextA046388[n_] := Block[{p1 = Prime[Range[2, PrimePi[Max[3, NextPrime[Ceiling@Sqrt[n + 1] - 1]]]]], p2}, p2 = Table[Max[NextPrime[p1[[i]]], NextPrime[Ceiling[(n + 1)/p1[[i]]] - 1]], {i, Length[p1]}]; Min[p1*p2]]; Table[seed=1; While[seed = NextA046388[seed]; fct = FactorInteger[seed]; p1 = fct[[1, 1]]; p2 = fct[[2, 1]]; c1 = 2^i*p1 + p2; c2 = p1 + 2^i*p2; ! ((PrimeQ[c1]) && (PrimeQ[c2]))]; seed, {i, 1, 56}]

%t nn = 2000; pq = Select[Range[nn], PrimeOmega[#] == 2 &]; p = Table[FactorInteger[r][[1, 1]], {r, pq}]; q = pq/p; t = {}; n = 1; While[i = 1; While[i <= Length[pq] && ! (PrimeQ[2^n*p[[i]] + q[[i]]] && PrimeQ[2^n*q[[i]] + p[[i]]]), i++]; i <= Length[pq], AppendTo[t, pq[[i]]]; n++]; t (* _T. D. Noe_, May 21 2013 *)

%Y Cf. A225916.

%K nonn

%O 1,1

%A _Lei Zhou_, May 20 2013