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%I #88 Mar 16 2016 11:19:43
%S 1,2,3,7,14,28,57,113,227,437,834,1616,3143,6144,12036,23467,45713,
%T 89375,175722,346193,681828,1344815,2657630,5253615,10374965,20471599,
%U 40401901,79871358,158182869,313402574,620776183,1228390053,2430853614,4813878134,9550070608
%N a(n) = the number of ways that at least two distinct primes <= prime(n) sum to a prime.
%C From _Bob Selcoe_, Oct 02 2015: (Start)
%C Conjectures:
%C i. a(n) ~ 2*a(n-1);
%C ii. a(n) <= 2*a(n-1)+1, a(n) < 2*a(n-1) n>=11;
%C iii. As n increases to infinity, a(n)/a(n-1) generally increases toward approximately 2, though the limiting ratio must be < 2.
%C (End)
%C From _Alois P. Heinz_, Oct 02 2015: (Start)
%C a(333) = 2*a(332)+d where d = 608...358 is a 95-digit positive integer.
%C It is not true that "a(n)/a(n-1) generally increases"; see plot below.
%C (End)
%C From _Bob Selcoe_, Oct 20 2015: (Start)
%C The plot does, in fact, suggest that a(n)/a(n-1) "generally increases" (i.e., generally a(z*n)/a(z*n-1) > a(n)/a(n-1), when z is sufficiently large). In other words, the peaks and the troughs tend to increase, with peaks tending to be higher than next trough. This behavior is consistent with the observation about "general increase" in conjecture iii, as n increases to infinity. However, the plot does not show that generally a(n)/a(n-1) > a(n+1)/a(n), or that there can't be a maximum value of a(n)/a(n-1) after which all other values decrease (two completely different issues).
%C Conjecture ii is clearly false since a(n)/(n-1) is slightly > 2 for a few terms 11 <= n <= 400 (n = {333..340}), therefore weakening the observation in conjecture iii that the limiting ratio must be < 2. (End)
%H Alois P. Heinz, <a href="/A262765/b262765.txt">Table of n, a(n) for n = 2..1000</a>
%H Alois P. Heinz, <a href="/A262765/a262765_3.jpg">Plot of a(n+1)/a(n)</a>
%F a(n) = A071810(n) - n. - _Alois P. Heinz_, Oct 23 2015
%e a(5)=7; prime(5)=11: 2+3=5, 2+5=7; 2+11=13; 2+3+5+7=17; 3+5+11=19; 2+3+7+11=23; 5+7+11=23.
%p s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+ithprime(n)) end:
%p b:= proc(n, i, t) option remember; `if`(n=0,
%p `if`(t=0, 1, 0), `if`(i<1, 0, b(n, i-1, t) +(p->
%p `if`(p>n, 0, b(n-p, i-1, max(0, t-1))))(ithprime(i))))
%p end:
%p a:= n-> add(`if`(isprime(k), b(k, n, 2), 0), k=5..s(n)):
%p seq(a(n), n=2..36); # _Alois P. Heinz_, Oct 01 2015
%t Length@ Select[Total /@ ReplaceAll[Subsets[Prime@ Range@ #], {_} -> Nothing], PrimeQ] & /@ Range[2, 21] (* _Michael De Vlieger_, Oct 01 2015 *)
%Y Cf. A000040 (prime numbers), A007504, A071810.
%K nonn
%O 2,2
%A _Bob Selcoe_, Sep 30 2015
%E a(10)-a(21) from _Michael De Vlieger_, Oct 01 2015
%E a(22)-a(36) from _Alois P. Heinz_, Oct 01 2015
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