cp's OEIS Frontend

A136009 Primes p for which the least pentagonal number q=n(3n+/-1)/2 which makes p+q prime satisfies 2q > p.

%I A136009 #18 Jan 23 2019 10:32:01
%S A136009 3,7,13,19,23,37,43,73,103,233,359,503
%N A136009 Primes p for which the least pentagonal number q=n(3n+/-1)/2 which makes p+q prime satisfies 2q > p.
%C A136009 Is the list complete?
%C A136009 For each prime p, the author is looking for the least generalized pentagonal number (A001318) q such that p+q is prime. The list of such q begins with: 1, 2, 2, 12, 2, 40, 2, 12, 126, 2, 12, 22, 2, 40, 12. In this list, most q's are much smaller than their corresponding p. The current sequence gives the primes p for which q is greater than p/2. - _Michel Marcus_, Aug 21 2013
%C A136009 No more terms up to 10^10. Heuristics suggest there are no more terms. - _Charles R Greathouse IV_, Aug 21 2013
%H A136009 G. N. Watson, Math. Notes 2103. <a href="http://www.jstor.org/stable/3611334">A Curious Property of the Prime Number 503</a>, Math. Gaz. vol 33, no 306, Dec 1949, pp. 302-303.
%e A136009 For p=23, q=126. For p=503, q=590.
%e A136009 More examples:
%e A136009 p=3, q=2, p+q=5.
%e A136009 p=7, q=12, p+q=19.
%e A136009 p=13, q=40, p+q=53.
%e A136009 p=19, q=12, p+q=31.
%e A136009 p=23, q=126, p+q=149.
%e A136009 p=37, q=22, p+q=59.
%e A136009 p=43, q=40, p+q=83.
%e A136009 p=73, q=40, p+q=113.
%e A136009 p=103, q=70, p+q=173.
%e A136009 p=233, q=126, p+q=359.
%e A136009 p=359, q=210, p+q=569.
%e A136009 p=503, q=590, p+q=1093. - _Michel Marcus_, Aug 21 2013
%o A136009 (PARI) isok(p) = my(nn = 1, ok = 0); until (ok, q = nn*(3*nn - 1)/2; if (isprime(p+q), ok = 1; if (2*q > p, return (q))); q = nn*(3*nn + 1)/2; if (isprime(p+q), ok = 1; if (2*q > p, return (q));); nn++;); 0 \\ _Michel Marcus_, Aug 21 2013
%o A136009 (PARI) is(p)=my(n=1,t);while(!isprime(t=p+n*(3*n-1)/2)&&!isprime(t=p+n*(3*n+1)/2)&&2*t<=3*p,n++); 2*t>3*p && isprime(p) \\ _Charles R Greathouse IV_, Aug 21 2013
%K A136009 more,nonn
%O A136009 1,1
%A A136009 _Chris K. Caldwell_, Mar 18 2008
%E A136009 a(10) = 233 inserted by _Michel Marcus_, Aug 21 2013