A009190 "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.

7310131732015251470110369, 2061519317176132799110061, 3756800873017263196139951, 6316254452384500173544921, 14733008374252975993271023, 532045863029111910656094569, 476592878661578285779747459

Offset: 20

Author

David W. Wilson

Keywords

• nonn
• more

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

• Jinyuan Wang, C++ program for computing the initial terms
• Eric Weisstein's World of Mathematics, Twin peaks

Crossrefs

Cf. A020639 (lpf), A058989.

Programs

PARI

is_TwinPeak(x)={forstep(k=2,2*p=factor(x)[1,1],2,factor(x+k,p)[1,1]
M. F. Hasler, Jan 28 2014

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

For n > 1, a(n) = -1 if A058989(n-1) < 2*prime(n) - 1. - Jinyuan Wang, Nov 27 2024

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

a(24)-a(26) from Jinyuan Wang, Nov 27 2024