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 A205558 (A204898)/2 = (prime(k)-prime(j))/2; A086802 without its zeros. 58
 1, 2, 1, 4, 3, 2, 5, 4, 3, 1, 7, 6, 5, 3, 2, 8, 7, 6, 4, 3, 1, 10, 9, 8, 6, 5, 3, 2, 13, 12, 11, 9, 8, 6, 5, 3, 14, 13, 12, 10, 9, 7, 6, 4, 1, 17, 16, 15, 13, 12, 10, 9, 7, 4, 3, 19, 18, 17, 15, 14, 12, 11, 9, 6, 5, 2, 20, 19, 18, 16, 15, 13, 12, 10, 7, 6, 3, 1, 22, 21 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let p(n) denote the n-th prime. If c is a positive integer, there are infinitely many pairs (k,j) such that c divides p(k)-p(j). The set of differences p(k)-p(j) is ordered as a sequence at A204890. Guide to related sequences: c....k..........j..........p(k)-p(j).[p(k)-p(j)]/c 2....A133196....A131818....A204898....A205558 3....A205560....A205547....A205557....A205675 4....A205677....A205678....A205681....A205682 5....A205684....A205685....A205688....A205689 6....A205691....A205692....A205695....A205696 7....A205698....A205699....A205702....A205703 8....A205705....A205706....A205709....A205710 9....A205712....A205713....A205716....A205717 10...A205720....A205721....A205724....A205725 It appears that, as rectangular array, this sequence can be described by A(n,k) is the least m such that there are k primes in the set prime(n) + 2*i for {i=1..n}. - Michel Marcus, Mar 29 2023 LINKS EXAMPLE Writing prime(k) as p(k), p(3)-p(2)=5-3=2 p(4)-p(2)=7-3=4 p(4)-p(3)=7-5=2 p(5)-p(2)=11-3=8 p(5)-p(3)=11-5=6 p(5)-p(4)=11-7=4, so that the first 6 terms of A205558 are 1,2,1,4,3,2. The sequence can be regarded as a rectangular array in which row n is given by [prime(n+2+k)-prime(n+1)]/2; a northwest corner follows: 1...2...4...5...7...8....10...13...14...17...19...20 1...3...4...6...7...9....12...13...16...18...19...21 2...3...5...6...8...11...12...15...17...18...20...23 1...3...4...6...9...10...13...15...16...18...21...24 2...3...5...8...9...12...14...15...17...20...23...24 1...3...6...7...10..12...13...15...18...21...22...25 2...5...6...9...11..12...14...17...20...21...24...26 - Clark Kimberling, Sep 29 2013 MATHEMATICA s[n_] := s[n] = Prime[n]; z1 = 200; z2 = 80; f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2]; Table[s[n], {n, 1, 30}] (* A000040 *) u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]] Table[u[m], {m, 1, z1}] (* A204890 *) v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0] w[n_] := w[n] = Table[v[n, h], {h, 1, z1}] d[n_] := d[n] = Delete[w[n], Position[w[n], 0]] c = 2; t = d[c] (* A080036 *) k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2] j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2 Table[k[n], {n, 1, z2}] (* A133196 *) Table[j[n], {n, 1, z2}] (* A131818 *) Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A204898 *) Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205558 *) CROSSREFS Cf. A205675, A205560, A204892. Sequence in context: A087850 A087849 A075015 * A082494 A194187 A174375 Adjacent sequences: A205555 A205556 A205557 * A205559 A205560 A205561 KEYWORD nonn,changed AUTHOR Clark Kimberling, Jan 30 2012 STATUS approved

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Last modified March 29 20:41 EDT 2023. Contains 361599 sequences. (Running on oeis4.)