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 A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral). 14
 83, 101, 127, 137, 163, 199, 233, 311, 373, 443, 463, 491, 541, 587, 613, 631, 641, 659, 673, 683, 691, 733, 757, 797, 859, 881, 911, 919, 953, 971, 991, 1013, 1051, 1061, 1103, 1109, 1117, 1193, 1201, 1213, 1249, 1307, 1319, 1409, 1433, 1459, 1483, 1487 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Isolated prime numbers have no adjacent primes in a lattice generated by writing consecutive integers starting from 1 in a spiral distribution. If n0 is the number of isolated primes and p the number of primes less than N, the ratio n0/p approaches 1 as N increases. If n1, n2, n3, n4 denote the number of primes with respectively 1, 2, 3, 4 adjacent primes in the lattice, the ratios n1/n0, n2/n1, n3/n2, n4/n3 approach 0 as N increases. The limits stand for any 2D lattice of integers generated by a priori criteria (i.e., not knowing distributions of primes) as Ulam's lattice. REFERENCES G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 22. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..10000 Eric Weisstein's World of Mathematics, Prime Spiral. EXAMPLE 83 is an isolated prime as the adjacent numbers in lattice 50, 51, 81, 82, 84, 123, 124, 125 are not primes. From Michael De Vlieger, Dec 22 2015: (Start) Spiral including n <= 17^2 showing only primes, with the isolated primes in parentheses (redrawn by Jon E. Schoenfield, Aug 06 2017):   257 .  .  .  .  . 251 .  .  .  .  .  .  .  .  . 241    . 197 .  .  . 193 . 191 .  .  .  .  .  .  .  .  .    .  .  .  .  .  .  .  . 139 .(137).  .  .  .  . 239    .(199).(101).  .  . 97  .  .  .  .  .  .  . 181 .    .  .  .  .  .  .  .  . 61  . 59  .  .  . 131 .  .    .  .  . 103 . 37  .  .  .  .  . 31  . 89  . 179 .   263 . 149 . 67  . 17  .  .  . 13  .  .  .  .  .  .    .  .  .  .  .  .  .  5  .  3  . 29  .  .  .  .  .    .  . 151 .  .  . 19  .  .  2 11  . 53  .(127).(233)    .  .  . 107 . 41  .  7  .  .  .  .  .  .  .  .  .    .  .  .  . 71  .  .  . 23  .  .  .  .  .  .  .  .    .  .  . 109 . 43  .  .  . 47  .  .  .(83) . 173 .   269 .  .  . 73  .  .  .  .  . 79  .  .  .  .  . 229    .  .  .  .  . 113 .  .  .  .  .  .  .  .  .  .  .   271 . 157 .  .  .  .  .(163).  .  . 167 .  .  . 227    . 211 .  .  .  .  .  .  .  .  .  .  . 223 .  .  .    .  .  .  . 277 .  .  . 281 . 283 .  .  .  .  .  . MAPLE # A is Ulam's lattice if (isprime(A[x, y])and(not(isprime(A[x+1, y]) or isprime(A[x-1, y])or isprime(A[x, y+1])or isprime(A[x, y-1])or isprime(A[x-1, y-1])or isprime(A[x+1, y+1])or isprime(A[x+1, y-1])or isprime(A[x-1, y+1])))) then print (A[x, y]) ; fi; MATHEMATICA spiral[n_] := Block[{o = 2 n - 1, t, w}, t = Table[0, {o}, {o}]; t = ReplacePart[t, {n, n} -> 1]; Do[w = Partition[Range[(2 (# - 1) - 1)^2 + 1, (2 # - 1)^2], 2 (# - 1)] &@ k; Do[t = ReplacePart[t, {(n + k) - (j + 1), n + (k - 1)} -> #[[1, j]]]; t = ReplacePart[t, {n - (k - 1), (n + k) - (j + 1)} -> #[[2, j]]]; t = ReplacePart[t, {(n - k) + (j + 1), n - (k - 1)} -> #[[3, j]]]; t = ReplacePart[t, {n + (k - 1), (n - k) + (j + 1)} -> #[[4, j]]], {j, 2 (k - 1)}] &@ w, {k, 2, n}]; t]; f[w_] := Block[{d = Dimensions@ w, t, g}, t = Reap[Do[Sow@ Take[#[[k]], {2, First@ d - 1}], {k, 2, Last@ d - 1}]][[-1, 1]] &@ w; g[n_] := If[n != 0, Total@ Join[Take[w[[Last@ # - 1]], {First@ # - 1, First@ # + 1}], {First@ #, Last@ #} &@ Take[w[[Last@ #]], {First@ # - 1, First@ # + 1}], Take[w[[Last@ # + 1]], {First@ # - 1, First@# + 1}]] &@(Reverse@ First@ Position[t, n] + {1, 1}) == 0, False]; Select[Union@ Flatten@ t, g@ # &]]; f[spiral@ 21 /. n_ /; CompositeQ@ n -> 0] (* Michael De Vlieger, Dec 22 2015, Version 10 *) CROSSREFS Cf. A001107, A002939, A007742, A033951-A033954, A033989, A033990, A033991, A002943, A033996, A033988, A014848. Cf. A113688 (isolated semiprimes in the semiprime spiral), A156859. Sequence in context: A335916 A158719 A160028 * A316970 A139965 A180523 Adjacent sequences:  A115255 A115256 A115257 * A115259 A115260 A115261 KEYWORD nonn AUTHOR Giorgio Balzarotti and Paolo P. Lava, Feb 17 2006 STATUS approved

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)