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a(1) is 4. a(n) is the least semiprime q (A001358) greater than p = a(n-1), such that p/q is a new minimum.
1

%I #22 Mar 03 2024 10:14:53

%S 4,6,10,21,46,106,247,579,1363,3214,7586,17915,42311,99931,236023,

%T 557455,1316638,3109733,7344803,17347513,40972678,96772393,228564417,

%U 539840885,1275037411,3011480697,7112745019,16799424206,39678162637,93714913738,221343037931,522784885426,1234753254431

%N a(1) is 4. a(n) is the least semiprime q (A001358) greater than p = a(n-1), such that p/q is a new minimum.

%C Inspired by and analogous to A265418.

%C p/q -> 0.423392190744304142156851442297311481582158896664...

%H Robert G. Wilson v, <a href="/A277343/b277343.txt">Table of n, a(n) for n = 1..200</a>

%e 4/6 is 0.666... is a new low or minimum;

%e 6/9 is 0.666... is not a new minimum, but;

%e 6/10 is 0.600... is a new minimum;

%e 10/21 is 0.476... is a new minimum;

%e 21/46 is 0.456... is a new minimum;

%e ... 522784885426/1234753254431 is 0.423... is a new minimum; etc.

%t NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]];

%t p = 4; q = 6; mx = 1; lst = {}; While[q < 10^15, r = p/q; If[r < mx, mx = r; AppendTo[lst, p]; p = q]; q = NextSemiPrime[Floor[q/r]]]; lst (* or *)

%t f[lst_List] := Block[{p = lst[[-2]], q = lst[[-1]]}, Append[lst, NextSemiPrime[ Floor[q^2/p]]]]; lst = {4, 6}; lst = Nest[f, lst, 30]

%Y Cf. A001358, A265418.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Oct 09 2016