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
KEYWORD
nonn
AUTHOR
Giorgio Balzarotti and Paolo P. Lava, Feb 17 2006
STATUS
approved