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A329405
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Among the pairwise sums of any three consecutive terms there is no prime: lexicographically earliest such sequence of distinct positive integers.
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28
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1, 3, 5, 7, 9, 11, 13, 14, 2, 4, 6, 8, 10, 12, 15, 18, 17, 16, 19, 20, 25, 24, 21, 27, 23, 22, 26, 28, 29, 34, 31, 32, 33, 30, 35, 39, 37, 38, 40, 36, 41, 44, 43, 42, 45, 46, 47, 48, 51, 54, 57, 58, 53, 52, 59, 56, 49, 50, 55, 60, 61, 62, 63, 66, 67, 68, 65, 64, 69, 71, 72, 70, 73, 74, 79, 76, 77, 78
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OFFSET
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1,2
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COMMENTS
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Conjectured to be a permutation of the positive integers.
Equivalently: For any n, neither a(n) + a(n+1) nor a(n) + a(n+2) is prime. Or: For any n and 0 <= i < j <= 2, a(n+i) + a(n+j) is never prime.
See A329450, A329452 onward and the wiki page for variants and further considerations about existence, surjectivity, etc. of such sequences. (End)
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LINKS
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EXAMPLE
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a(1) = 1 from minimality.
a(2) = 3 since 2 would produce 3 (a prime) by making 1 + 2.
a(3) = 5 since 2 or 4 would produce a prime (e.g., 3 + 4 = 7).
a(4) = 7 since 2, 4 or 6 would produce a prime (e.g., 5 + 6 = 11).
...
a(8) = 14 as 2, 4, 6, 8, 10 or 12 would produce a prime together with a(7) = 13 or a(6) = 11.
a(9) = 2 as neither 2 + 13 = 15 nor 2 + 14 = 16 is prime.
And so on.
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MATHEMATICA
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a[1]=1; a[2]=3; a[n_]:=a[n]=(k=1; While[Or@@PrimeQ[Plus@@@Subsets[{a[n-1], a[n-2], ++k}, {2}]]||MemberQ[Array[a, n-1], k]]; k); Array[a, 100] (* Giorgos Kalogeropoulos, May 09 2021 *)
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PROG
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(PARI) A329405(n, show=1, o=1, p=o, U=[])={for(n=o, n-1, show&&print1(p", "); U=setunion(U, [p]); while(#U>1&&U[1]==U[2]-1, U=U[^1]); for(k=U[1]+1, oo, setsearch(U, k) || isprime(o+k) || isprime(p+k) || [o=p, p=k, break])); p} \\ Optional args: show=0: don't print the list; o=0: start with a(0) = 0, i.e., compute A329450. See the wiki page for more general code returning a vector: S(n, 0, 3, 1) = a(1..n).
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CROSSREFS
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Cf. A329333 (3 consecutive terms, exactly 1 prime sum).
Cf. A329406 .. A329410 (exactly 1 prime sum using 4, ..., 10 consecutive terms).
Cf. A329411 .. A329416 (exactly 2 prime sums using 3, ..., 10 consecutive terms).
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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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