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A009190
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"2p twin peaks": a(n) = least x with lpf(x) = lpf(x + 2p) = p = prime(n) and lpf(y) < p for all x < y < x + 2p, where lpf = least prime factor; a(p) = -1 if no such x exists.
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1
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OFFSET
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20,1
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COMMENTS
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For prime p, a 2p-twin peak is a number x such that lpf(x) = lpf(x+2p) = p and x < y < x+2p => lpf(y) < p. (lpf(n) = least prime factor of n.) p = 71 is the smallest prime admitting a 2p-twin peak.
a(30) <= 126972592296404970720882679404584182254788131, found by Manjul Bhargava, John Conway, Johan de Jong, and Derek Smith in 1997. - Mauro Fiorentini, Feb 16 2020 [Comment corrected by N. J. A. Sloane, Aug 19 2020: they found an upper bound on a(30), they did not prove it is equal to a(30). A typo in the names has also been corrected.]
a(1), ..., a(13) = -1, as the sequence of integers with small least prime factor is not long enough, see A058989; the minimum prime for which twin peaks exist is between 43 (a(14)) and 71 (a(20)). - Mauro Fiorentini, Feb 17 2020
It is (weakly) conjectured that a(n) = -1 for 14 <= n <= 19, which is why this entry has offset 20. - N. J. A. Sloane, Aug 19 2020
a(24)=9503844926749390990454854843625839 was incorrect. There exist smaller 2p-twin peaks for p=prime(24)=89, such as 71945201112472689127120879, which is an upper bound for a(24).
a(25) <= 168113372406632936032276646039033.
a(26) <= 91655763448408439742416249179.
a(27) <= 3295708850046747547035632762993. (End)
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REFERENCES
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Various postings to the Math-Fun mailing list, 1996-1997. See in particular the post by David W. Wilson on Feb 10 1997.
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LINKS
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FORMULA
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Assuming a(n) > 0, then a(n) < A002110(n)/2, since if (x,x+2p) is a 2p-twin peak, then so is (q-x-2p,q-x), where q=A034386(p). - M. F. Hasler, Jan 28 2014
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PROG
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(PARI) is_TwinPeak(x)={forstep(k=2, 2*p=factor(x)[1, 1], 2, factor(x+k, p)[1, 1]<p || return(k==2*p))} \\ M. F. Hasler, Jan 28 2014
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Aug 19 2020, based in part on email correspondence with Manjul Bhargava.
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STATUS
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approved
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