

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




OFFSET

20,1


COMMENTS

For prime p, a 2ptwin 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 2ptwin 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 2ptwin 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 MathFun mailing list, 19961997. See in particular the post by David W. Wilson on Feb 10 1997.


LINKS



FORMULA

Assuming a(n) > 0, then a(n) < A002110(n)/2, since if (x,x+2p) is a 2ptwin peak, then so is (qx2p,qx), 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



KEYWORD

nonn,more


AUTHOR



EXTENSIONS

Entry revised by N. J. A. Sloane, Aug 19 2020, based in part on email correspondence with Manjul Bhargava.


STATUS

approved



