A217016 - OEIS

%I #23 Nov 13 2019 14:59:45

%S 8,24,90,210,840,10920,13650,39270,1492260,11741730,281291010,

%T 10919808900

%N Least r > 1 without Goldbach partition 2r = p+q such that |p-q| is prime(n)-smooth.

%C More explicitly, a(n) is the smallest r > 1 such that, whenever 2r is the sum of two primes, their difference has a prime factor larger than prime(n).

%C Conjecture [D. Broadhurst]: a(n) is prime(n)-smooth.

%C a(11=primepi(31)) <= 281291010 = 29#/23, where P# = A034386(P). [This limit is quickly found using the PARI code A217016_bound(11,3e8).]

%C a(12=primepi(37)) <= 10919808900 = 30*17#*23*31. [D. Broadhurst]

%H D. Broadhurst (in reply to M. Underwood), <a href="http://groups.yahoo.com/group/primenumbers/message/24482">Re: Here's some Goldbach separation data</a>, Yahoo! group "primenumbers", Sep 23 2012.

%H Warren Smith and others, <a href="/A217016/a217016.txt">Goldbach separation data</a>, digest of 18 messages in primenumbers Yahoo group, Sep 18 - Sep 25, 2012 (initial sequence terms provided in message 6).

%o (PARI) A217016(n,m=2,M=9e99)={my(p=prime(n),/* find a p-smooth Goldbach partition: */ sG(N,p)=forprime(q=1,N\2,isprime(N-q)||next; vecmax(factor(N-2*q,p)[,1])>p||return(q))); /* main prog: */ forstep(N=m*2,M,2, sG(N,p)||return(N\2))} /* This brute force approach becomes too slow for n > primepi(19). */

%o (PARI) A217016_bound(n,B/*upper bound*/,m/*lower bound*/,verbose=1)={my(p=prime(n),P=1,sG(N,p)=forprime(q=1,N\2,isprime(N-q)||next; vecmax(factor(N-2*q,p)[,1])>p||return(q))); /*init: default value for B & m*/ B || B=prod(i=1,n,prime(i),prime(n\2+1)); m || m=B\1.5; /*main*/ forprime(q=1,p,P<B||break;P*=q); my(F=factor(P)[,1]); forstep(i=#F,1,-1,my(N=P/F[i]); N<B||break; verbose & print1([[i]]); for(k=max(m\N,1),B\N,verbose & print1([k]);sG(2*k*N,p)||return(k*N)))}

%K nonn,more,hard

%O 1,1

%A _David Broadhurst_ and _M. F. Hasler_, Sep 23 2012

%E Terms a(11)-a(12) from _Robert Gerbicz_, Sep 25 2012.