A009190 - OEIS

%I #58 Nov 29 2024 08:09:49

%S 7310131732015251470110369,2061519317176132799110061,

%T 3756800873017263196139951,6316254452384500173544921,

%U 14733008374252975993271023,532045863029111910656094569,476592878661578285779747459

%N "2p-twin peaks": a(n) is the 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(n) = -1 if no such x exists.

%C 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.

%C 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.]

%C 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

%C 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

%C From _Brian Kehrig_, May 23 2023: (Start)

%C 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).

%C a(25) <= 168113372406632936032276646039033.

%C a(26) <= 91655763448408439742416249179.

%C a(27) <= 3295708850046747547035632762993. (End)

%D Various postings to the Math-Fun mailing list, 1996-1997. See in particular the post by David W. Wilson on Feb 10 1997.

%H Jinyuan Wang, <a href="/A009190/a009190.cpp.txt">C++ program for computing the initial terms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPeaks.html">Twin peaks</a>

%F 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

%F For n > 1, a(n) = -1 if A058989(n-1) < 2*prime(n) - 1. - _Jinyuan Wang_, Nov 27 2024

%o (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

%Y Cf. A020639 (lpf), A058989.

%K nonn,more

%O 20,1

%A _David W. Wilson_

%E a(24), found by Fred Helenius, added by _Mauro Fiorentini_, Feb 16 2020

%E Entry revised by _N. J. A. Sloane_, Aug 19 2020, based in part on email correspondence with Manjul Bhargava.

%E Incorrect a(24) removed by _Brian Kehrig_, May 23 2023

%E a(24)-a(26) from _Jinyuan Wang_, Nov 27 2024