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

1, 2, 3, 7, 14, 28, 57, 113, 227, 437, 834, 1616, 3143, 6144, 12036, 23467, 45713, 89375, 175722, 346193, 681828, 1344815, 2657630, 5253615, 10374965, 20471599, 40401901, 79871358, 158182869, 313402574, 620776183, 1228390053, 2430853614, 4813878134, 9550070608

Offset: 2

Author

Bob Selcoe, Sep 30 2015

Keywords

• nonn

Comments

From Bob Selcoe, Oct 02 2015: (Start)

Conjectures:

i. a(n) ~ 2*a(n-1);

ii. a(n) <= 2*a(n-1)+1, a(n) < 2*a(n-1) n>=11;

iii. As n increases to infinity, a(n)/a(n-1) generally increases toward approximately 2, though the limiting ratio must be < 2.

(End)

From Alois P. Heinz, Oct 02 2015: (Start)

a(333) = 2*a(332)+d where d = 608...358 is a 95-digit positive integer.

It is not true that "a(n)/a(n-1) generally increases"; see plot below.

(End)

From Bob Selcoe, Oct 20 2015: (Start)

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).

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)

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000
• Alois P. Heinz, Plot of a(n+1)/a(n)

Crossrefs

Cf. A000040 (prime numbers), A007504, A071810.

Programs

Maple

s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+ithprime(n)) end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1, 0, b(n, i-1, t) +(p->
      `if`(p>n, 0, b(n-p, i-1, max(0, t-1))))(ithprime(i))))
    end:
a:= n-> add(`if`(isprime(k), b(k, n, 2), 0), k=5..s(n)):
seq(a(n), n=2..36);  # Alois P. Heinz, Oct 01 2015

Mathematica

Length@ Select[Total /@ ReplaceAll[Subsets[Prime@ Range@ #], {} -> Nothing], PrimeQ] & /@ Range[2, 21] (* _Michael De Vlieger, Oct 01 2015 *)

Formula

a(n) = A071810(n) - n. - Alois P. Heinz, Oct 23 2015

Extensions

a(10)-a(21) from Michael De Vlieger, Oct 01 2015

a(22)-a(36) from Alois P. Heinz, Oct 01 2015