cp's OEIS Frontend

A176955 Primes p such that q=3//p, r=p//3, R(q) and R(r) are primes.

%I A176955 #1 Jun 01 2010 03:00:00
%S A176955 7,11,37,73,191,373,719,929,1033,1193,3301,3461,3911,3931,10223,10771,
%T A176955 12071,12451,13669,13931
%N A176955 Primes p such that q=3//p, r=p//3, R(q) and R(r) are primes.
%C A176955 q and r are emirps (see A006567) as R(q) and R(r) are requested to be primes too
%C A176955 3 = prime(2) is first prime for such a "construction" with prefixed/appended number and reversals
%C A176955 Necessarily FSD (First Significant Digit) of p is 1, 3, 7 or 9
%C A176955 If such a p is a palindromic prime, i.e. p = R(p), then q = R(r) and r = R(q):
%C A176955 Proof: q = 3//p = 3//R(p) = R(p//3) = R(r), r = p//3 = R(p)//3 = R(3//p) = R(q).
%C A176955 If such a p is an emirp, i.e. R(p) also prime, then R(p) is also term of sequence:
%C A176955 Proof: 3//R(p) = R(p//3) = R(r), R(p)//3 = R(3//p) = R(q), R(3//R(p)) = p//3 = r, R(R(p)//3) = 3//p = q.
%C A176955 List of (p: q, r, R(q), R(r))
%C A176955 (7=palprime(4): 37, 73, 73, 37), (11=palprime(5): 311, 113, 113, 311),
%C A176955 (37=emirp(4): 337, 373, 733, 373), (73=emirp(6): 373, 733, 373, 337),
%C A176955 (191=palprime(10): 3191, 1913, 1913, 3191), (373=palprime(13): 3373, 3733, 3733, 3373),
%C A176955 (719=prime(128): 3719, 7193, 9173, 3917), (929=palprime(20): 3929, 9293, 9293, 3929),
%C A176955 (1033=emirp(40): 31033, 10333, 33013, 33301), (1193=emirp(50): 31193, 11933, 39113, 33911),
%C A176955 (3301=emirp(115): 33301, 33013, 10333, 31033), (3461=prime(484): 33461, 34613, 16433, 31643),
%C A176955 (3911=emirp(145): 33911, 39113, 11933, 31193), (3931=prime(546): 33931, 39313, 13933, 31393),
%C A176955 (10223=prime(1254): 310223, 102233, 322013, 332201), (10771=prime(1312): 310771, 107713, 177013, 317701),
%C A176955 (12071=emirp(326): 312071, 120713, 170213, 317021), (12451=prime(1486): 312451, 124513, 154213, 315421),
%C A176955 (13669=prime(1614): 313669, 136693, 966313, 396631), (13931=palprime(31): 313931, 139313, 139313, 313931)
%D A176955 Peter Bundschuh: Einfuehrung in die Zahlentheorie, 6. Auflage, Springer, Berlin, 2008
%D A176955 Martin Gardner: Die magischen Zahlen des Dr. Matrix , Wolfgang Krueger Verlag FrankfurtMain, 1987
%e A176955 q=3//7=37=prime(12), r=7//3=73=prime(21), R(q)=r, R(r)=q, p=7=prime(4) is 1st term
%e A176955 q=3//719=3719=prime(519), r=719//3=7193=prime(919), R(q)=9173=prime(1137), R(r)=3917=prime(452),
%e A176955 p=719=prime(2^7) is 7th term and the first where q, r, R(q), R(r) are four different primes
%Y A176955 Cf. A000040, A006567, A176316, A176601, A176838
%K A176955 base,nonn,uned
%O A176955 1,1
%A A176955 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 29 2010