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 A217016 Least r > 1 without Goldbach partition 2r = p+q such that |p-q| is prime(n)-smooth. 2
 8, 24, 90, 210, 840, 10920, 13650, 39270, 1492260, 11741730, 281291010, 10919808900 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). Conjecture [D. Broadhurst]: a(n) is prime(n)-smooth. a(11=primepi(31)) <= 281291010 = 29#/23, where P# = A034386(P). [This limit is quickly found using the PARI code A217016_bound(11,3e8).] a(12=primepi(37)) <= 10919808900 = 30*17#*23*31. [D. Broadhurst] LINKS D. Broadhurst (in reply to M. Underwood), Re: Here's some Goldbach separation data, Yahoo! group "primenumbers", Sep 23 2012. Warren Smith and others, Goldbach separation data, digest of 18 messages in primenumbers Yahoo group, Sep 18 - Sep 25, 2012 (initial sequence terms provided in message 6). PROG (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). */ (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

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Last modified June 18 23:58 EDT 2021. Contains 345125 sequences. (Running on oeis4.)