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A115258
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Isolated primes in Ulam's lattice (1, 2, ... in spiral).
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15
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 22.
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LINKS
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EXAMPLE
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83 is an isolated prime as the adjacent numbers in lattice 50, 51, 81, 82, 84, 123, 124, 125 are not primes.
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 . . . . . .
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MAPLE
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# 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;
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A001107, A002939, A007742, A033951-A033954, A033989, A033990, A033991, A002943, A033996, A033988, A014848.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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