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A372263
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Least odd prime factor of the n-th sum of two consecutive primes, A001043(n) = prime(n) + prime(n+1), or 2 if there is no odd prime factor.
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0
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5, 2, 3, 3, 3, 3, 3, 3, 13, 3, 17, 3, 3, 3, 5, 7, 3, 2, 3, 3, 19, 3, 43, 3, 3, 3, 3, 3, 3, 3, 3, 67, 3, 3, 3, 7, 5, 3, 5, 11, 3, 3, 3, 3, 3, 5, 7, 3, 3, 3, 59, 3, 3, 127, 5, 7, 3, 137, 3, 3, 3, 3, 3, 3, 3, 3, 167, 3, 3, 3, 89, 3, 5, 47, 3, 193, 3, 3, 3, 3, 3, 3, 3, 109, 3, 223
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OFFSET
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1,1
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COMMENTS
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Since the sum of any two primes > 2 is even, we rather consider odd prime factors.
Can it be proved or disproved that there are primes that occur only finitely many times (or never) in this sequence? If so, which is the smallest such prime?
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LINKS
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FORMULA
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EXAMPLE
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Sums of two consecutive primes are given as s(n) = A001043(n). The least odd prime factor (or 2 if there's no odd prime factor) of these terms is a(n):
n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
s = 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, ...
a = 5, 2, 3, 3, 3, 3, 3, 3, 13, 3, 17, 3, 3, 3, 5, 7, 3, 2, ...
Also, a(21) = spf(152) = 19; a(23) = spf(172) = 43; a(32) = spf(268) = 67, ...
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PROG
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/* a "self-contained" but less efficient definition:
a(n) = factor(max((n=prime(n)+prime(n+1))>>valuation(n, 2), 2))[1, 1] */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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