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A132470
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Smallest number at distance exactly 3n from nearest prime.
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3
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2, 26, 119, 532, 1339, 1342, 9569, 15704, 19633, 31424, 31427, 31430, 31433, 155960, 155963, 360698, 360701, 370312, 370315, 492170, 1357261, 1357264, 1357267, 2010802, 2010805, 4652428, 17051785, 17051788, 17051791, 17051794, 17051797, 20831416, 20831419, 20831422
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OFFSET
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0,1
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COMMENTS
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Let f(m)= A051699(m) = exact distance from m to its closest prime (including m itself). Then a(n) = min { m : f(m) = 3n}. - R. J. Mathar, Nov 18 2007
This sequence can be derived from the record prime gap sequences A002386 and A005250. In particular, for n > 0, a(n) = A002386(k) + 3*n where k is the least index such that A005250(k) >= 3*n. - Andrew Howroyd, Jan 04 2020
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LINKS
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FORMULA
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EXAMPLE
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a(3)=532 where 532+3*3 is prime and all numbers below 532 have a distance smaller or larger than 3n=9 to their nearest primes and there is no prime within a distance of 8 to 532.
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MAPLE
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A051699 := proc(m) if isprime(m) then 0 ; elif m <= 2 then op(m+1, [2, 1]) ; else min(nextprime(m)-m, m-prevprime(m)) ; fi ; end: A132470 := proc(n) local m ; if n = 0 then RETURN(2); else for m from 0 do if A051699(m) = 3 * n then RETURN(m) ; fi ; od: fi ; end: seq(A132470(n), n=0..18) ; # R. J. Mathar, Nov 18 2007
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MATHEMATICA
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terms = 34;
gaps = Cases[Import["https://oeis.org/A002386/b002386.txt", "Table"], {_, _}][[;; terms, 2]];
w[n_] := (NextPrime[gaps[[n]] + 1] - gaps[[n]])/6 // Floor;
k = 1; a[0] = 2;
For[n = 1, n <= terms, n++, While[w[k] < n, k++]; a[n] = gaps[[k]] + 3n];
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PROG
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(PARI) \\ here R(gaps) wants prefix of A002386 as vector.
aA002386(lim)={my(L=List(), q=2, g=0); forprime(p=3, lim, if(p-q>g, listput(L, q); g=p-q); q=p); Vec(L)}
R(gaps)={my(w=vector(#gaps, n, nextprime(gaps[n]+1) - gaps[n])\6, r=vector(w[#w]+1), k=1); r[1]=2; for(n=1, w[#w], while(w[k]<n, k++); r[n+1]=gaps[k] + 3*n); r}
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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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Both this sequence and A051728 should be checked. There are two possibilities for confusion in each case. In defining f(m), does one allow or exclude m itself, in case m is a prime? In defining a(n), does one require (here) that f(m) = 3n or only that >= 3n, or (in A051728) that f(m) = 2n or only >= 2n? Probably there should be several sequences, to include all the possibilities in each case. - N. J. A. Sloane, Nov 18 2007. Added Nov 20 2007: R. J. Mathar has now clarified the definition of the present sequence.
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STATUS
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approved
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