A132470 Smallest number at distance exactly 3n from nearest prime.

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

Offset: 0

Author

Jonathan Vos Post, Sep 03 2007

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..258

Crossrefs

Cf. A002386, A005250, A051699, A051728.

Programs

Maple

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

Mathematica

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];
a /@ Range[0, terms-1] (* Jean-François Alcover, Apr 09 2020, after Andrew Howroyd *)

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]A002386(10^7))} \\ Andrew Howroyd, Jan 04 2020

Formula

a(n) = min {m : A051699(m) = 3n}. - R. J. Mathar, Nov 18 2007

Extensions

Corrected by Dean Hickerson, Sep 05 2007

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.

Corrected and extended by R. J. Mathar, Nov 18 2007

Terms a(19) and beyond from Andrew Howroyd, Jan 04 2020