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A262765
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a(n) = the number of ways that at least two distinct primes <= prime(n) sum to a prime.
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2
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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
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OFFSET
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2,2
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COMMENTS
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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)
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)
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)
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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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)):
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MATHEMATICA
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Length@ Select[Total /@ ReplaceAll[Subsets[Prime@ Range@ #], {_} -> Nothing], PrimeQ] & /@ Range[2, 21] (* Michael De Vlieger, Oct 01 2015 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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