A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).

10, 26, 50, 170, 170, 362, 362, 842, 842, 1370, 1370, 1850, 1850, 2210, 3722, 3722, 3722, 4892, 5042, 7082, 7922, 7922, 7922, 10610, 10610, 10610, 11450, 13844, 16130, 16130, 17162, 19322, 19322, 24614, 24614, 25592, 29504, 29930, 29930, 36020, 36020

Offset

2,1

Comments

The sequence is connected with a sufficient condition for the existence of an infinity of twin primes. In contrast to A242489, this sequence is nondecreasing.

All even numbers of the form A062326(n)^2 + 1 are in the sequence. All a(n)-1 are semiprimes. - Vladimir Shevelev, May 24 2014

a(n) <= A242489(n); a(n) >= prime(n)^2+1. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014

Conjecture: all numbers a(n)-3 are primes. Peter J. C. Moses verified this conjecture up to a(2001) (cf. with conjecture in A242720). - Vladimir Shevelev, Jun 09 2014

Links

Jinyuan Wang, Table of n, a(n) for n = 2..5000 (terms 2..2001 from Peter J. C. Moses).

Formula

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246819). - Vladimir Shevelev, Sep 02 2014

a(n) = prime(n)^2 + 1 for and only for numbers n>=2 which are in A137291. - Vladimir Shevelev, Sep 04 2014

Mathematica

lpf[k_] := FactorInteger[k][[1, 1]];
a[n_] := a[n] = For[k = If[n == 2, 10, a[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
Array[a, 50, 2] (* Jean-François Alcover, Nov 06 2018 *)

Prog

(PARI)
lpf(k) = factorint(k)[1, 1];
vector(50, n, k=6; while(lpf(k-3)<=lpf(k-1) || lpf(k-1)<prime(n+1), k+=2); k) \\ Colin Barker, Jun 01 2014

Crossrefs

Cf. A001359, A006512, A062326, A137291, A242489, A242490, A242847, A243960, A245363, A246501, A246748, A246819, A247011.

Sequence in context: A137351 A134406 A099978 * A242489 A074789 A229308

Adjacent sequences: A242716 A242717 A242718 * A242720 A242721 A242722

Keyword

nonn

Author

Vladimir Shevelev, May 21 2014

Status

approved