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A217016
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Least r > 1 without Goldbach partition 2r = p+q such that |p-q| is prime(n)-smooth.
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2
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8, 24, 90, 210, 840, 10920, 13650, 39270, 1492260, 11741730, 281291010, 10919808900
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OFFSET
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1,1
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COMMENTS
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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]
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LINKS
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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).
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PROG
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(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<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)))}
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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