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 A009190 "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. 1
 7310131732015251470110369, 2061519317176132799110061, 3756800873017263196139951, 6316254452384500173544921 (list; graph; refs; listen; history; text; internal format)
 OFFSET 20,1 COMMENTS 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 From Brian Kehrig, May 23 2023: (Start) 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) REFERENCES Various postings to the Math-Fun mailing list, 1996-1997. See in particular the post by David W. Wilson on Feb 10 1997. LINKS Table of n, a(n) for n=20..23. Eric Weisstein's World of Mathematics, Twin peaks FORMULA 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 PROG (PARI) is_TwinPeak(x)={forstep(k=2, 2*p=factor(x)[1, 1], 2, factor(x+k, p)[1, 1]

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Last modified July 21 16:38 EDT 2024. Contains 374475 sequences. (Running on oeis4.)