login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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]<p || return(k==2*p))} \\ M. F. Hasler, Jan 28 2014
CROSSREFS
Cf. A020639 (lpf).
Sequence in context: A104267 A113538 A280347 * A095444 A309072 A217416
KEYWORD
nonn,more
AUTHOR
EXTENSIONS
a(24), found by Fred Helenius, added by Mauro Fiorentini, Feb 16 2020
Entry revised by N. J. A. Sloane, Aug 19 2020, based in part on email correspondence with Manjul Bhargava.
Incorrect a(24) removed by Brian Kehrig, May 23 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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