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 A283302 Maximum number of pairs of primes (p,q) such that p < q <= prime(n) and p + q = constant. 2
 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 8, 9, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 15, 16, 17, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 21, 21, 22, 23, 24, 24, 24, 24, 24, 24, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Maximum number of different ways of expressing a number as a sum of two distinct primes less than or equal to prime(n). Is there any n such that a(n+1) - a(n) > 1? What is the asymptotic behavior of a(n)? To answer the first question: for all n, either a(n+1) = a(n) or a(n+1) = a(n) + 1. - Charles R Greathouse IV, Mar 06 2017 LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 FORMULA a(n) >> n/log n. In particular, lim inf a(n) * (log n)/n >= 1/4. - Charles R Greathouse IV, Mar 06 2017 EXAMPLE a(1)=0 because there are no two distinct primes less than or equal to prime(1)=2. a(2)=1 because there are only two distinct primes less than or equal to prime(2)=3, and then there is only one sum among them: 2 + 3 = 5. a(3)=1 because the three pairs of distinct primes less than or equal to prime(3)=5, i.e., (2,3), (3,5), and (2,5), produce different sums: 2 + 3 = 5, 3 + 5 = 8, and 2 + 5 = 7. a(6)=2 because among all pairs of distinct primes taken from the first six primes, 2, 3, 5, 7, 11, and 13, there are at most two pairs with same sum, e.g., 3 + 13 = 5 + 11 = 16. a(8)=3 because among all pairs of distinct primes taken from the first eight primes, 2, 3, 5, 7, 11, 13, 17, and 19, there are at most three pairs with the same sum, i.e., 5 + 19 = 7 + 17 = 11 + 13 = 24. MATHEMATICA a[n_]:=Module[{fp, fps, fpst, fpstt, fpstts, fpsttst}, fp=Prime[Range[n]]; fps=Subsets[fp, {2}]; fpst=Table[Total[fps[[j]]], {j, 1, Length[fps]}]; fpstt=fpst//Tally; fpstts=fpstt//Sort[#, #1[[2]]>#2[[2]]&]&; If [n<2, 0, fpsttst=fpstts//Transpose; fpsttst[[2]]//Max]//Return]; Table[a[n], {n, 1, 120}] PROG (PARI) first(n)=my(v=vector(n), P=primes(n), H=vectorsmall((P[#P]+P[#P-1])/2)); v[2]=1; for(n=3, #P, for(i=2, n-1, H[(P[n]+P[i])/2]++); v[n]=vecmax(H)); v \\ Charles R Greathouse IV, Mar 06 2017 CROSSREFS Sequence in context: A084520 A084510 A053620 * A225593 A057360 A057364 Adjacent sequences:  A283299 A283300 A283301 * A283303 A283304 A283305 KEYWORD nonn AUTHOR Andres Cicuttin, Mar 04 2017 STATUS approved

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Last modified August 22 02:44 EDT 2019. Contains 326169 sequences. (Running on oeis4.)