cp's OEIS Frontend

A262765 a(n) = the number of ways that at least two distinct primes <= prime(n) sum to a prime.

%I A262765 #88 Mar 16 2016 11:19:43
%S A262765 1,2,3,7,14,28,57,113,227,437,834,1616,3143,6144,12036,23467,45713,
%T A262765 89375,175722,346193,681828,1344815,2657630,5253615,10374965,20471599,
%U A262765 40401901,79871358,158182869,313402574,620776183,1228390053,2430853614,4813878134,9550070608
%N A262765 a(n) = the number of ways that at least two distinct primes <= prime(n) sum to a prime.
%C A262765 From _Bob Selcoe_, Oct 02 2015: (Start)
%C A262765 Conjectures:
%C A262765 i. a(n) ~ 2*a(n-1);
%C A262765 ii. a(n) <= 2*a(n-1)+1, a(n) < 2*a(n-1) n>=11;
%C A262765 iii. As n increases to infinity, a(n)/a(n-1) generally increases toward approximately 2, though the limiting ratio must be < 2.
%C A262765 (End)
%C A262765 From _Alois P. Heinz_, Oct 02 2015: (Start)
%C A262765 a(333) = 2*a(332)+d where d = 608...358 is a 95-digit positive integer.
%C A262765 It is not true that "a(n)/a(n-1) generally increases"; see plot below.
%C A262765 (End)
%C A262765 From _Bob Selcoe_, Oct 20 2015: (Start)
%C A262765 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 A262765 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 A262765 Alois P. Heinz, <a href="/A262765/b262765.txt">Table of n, a(n) for n = 2..1000</a>
%H A262765 Alois P. Heinz, <a href="/A262765/a262765_3.jpg">Plot of a(n+1)/a(n)</a>
%F A262765 a(n) = A071810(n) - n. - _Alois P. Heinz_, Oct 23 2015
%e A262765 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 A262765 s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+ithprime(n)) end:
%p A262765 b:= proc(n, i, t) option remember; `if`(n=0,
%p A262765       `if`(t=0, 1, 0), `if`(i<1, 0, b(n, i-1, t) +(p->
%p A262765       `if`(p>n, 0, b(n-p, i-1, max(0, t-1))))(ithprime(i))))
%p A262765     end:
%p A262765 a:= n-> add(`if`(isprime(k), b(k, n, 2), 0), k=5..s(n)):
%p A262765 seq(a(n), n=2..36);  # _Alois P. Heinz_, Oct 01 2015
%t A262765 Length@ Select[Total /@ ReplaceAll[Subsets[Prime@ Range@ #], {_} -> Nothing], PrimeQ] & /@ Range[2, 21] (* _Michael De Vlieger_, Oct 01 2015 *)
%Y A262765 Cf. A000040 (prime numbers), A007504, A071810.
%K A262765 nonn
%O A262765 2,2
%A A262765 _Bob Selcoe_, Sep 30 2015
%E A262765 a(10)-a(21) from _Michael De Vlieger_, Oct 01 2015
%E A262765 a(22)-a(36) from _Alois P. Heinz_, Oct 01 2015