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A262765 a(n) = the number of ways that at least two distinct primes <= prime(n) sum to a prime. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
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)
LINKS
Alois P. Heinz, Plot of a(n+1)/a(n)
FORMULA
a(n) = A071810(n) - n. - Alois P. Heinz, Oct 23 2015
EXAMPLE
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.
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 *)
CROSSREFS
Cf. A000040 (prime numbers), A007504, A071810.
Sequence in context: A294627 A293326 A308092 * A340163 A131666 A135258
KEYWORD
nonn
AUTHOR
Bob Selcoe, Sep 30 2015
EXTENSIONS
a(10)-a(21) from Michael De Vlieger, Oct 01 2015
a(22)-a(36) from Alois P. Heinz, Oct 01 2015
STATUS
approved

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Last modified March 28 16:11 EDT 2024. Contains 371254 sequences. (Running on oeis4.)