A243937 - OEIS

%I #30 Sep 03 2014 10:29:22

%S 6,8,12,14,18,20,24,30,32,36,38,42,44,48,54,60,62,66,68,72,74,78,80,

%T 84,90,96,98,102,104,108,110,114,120,122,126,128,132,138,140,144,150,

%U 152,156,158,162,164,168,174,180,182,186,188,192,194,198,200,204,210

%N Even numbers n>=6 for which lpf(n-1) > lpf(n-3), where lpf = least prime factor.

%C Complement of A245024 over even n >= 6.

%C Conjecture: All differences are 2, 4 or 6 such that there are no two consecutive terms 2 (..., 2, 2, ...), no two consecutive terms 4, while consecutive terms 6 occur 1, 2, 3 or 4 times; also consecutive pairs of terms 2, 4 appear 1, 2, 3 or 4 times. The conjecture is verified up to n = 2.5*10^7. - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jul 11 2014

%C Divisibility by 3 means 6m is in the sequence for all m > 0, and 6m + 4 never is, while 6m + 2 is undetermined. Divisibility by 5 means 30m + 8 is always in the sequence, and 30m + 26 never is. This proves the above conjecture. - _Jens Kruse Andersen_, Aug 19 2014

%C Note that,

%C 1) Since numbers of the form 6*k evidently are in the sequence, then the counting function of the terms not exceeding x is not less than x/6.

%C 2) Sequence {a(n)-1} contains all primes greater than 3 in the natural order. The subsequence of other terms of {a(n)-1} is 35, 65, 77, 95, ... - _Vladimir Shevelev_, Jul 15 2014

%H Jens Kruse Andersen, <a href="/A243937/b243937.txt">Table of n, a(n) for n = 1..10000</a>

%o (PARI) select(n->factor(n-1)[1,1]>factor(n-3)[1,1], vector(200, x, 2*x+4)) \\ _Jens Kruse Andersen_, Aug 19 2014

%Y Cf. A242719, A242720, A245024.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Jul 10 2014

%E More terms from _Peter J. C. Moses_, Jul 10 2014