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Search: id:A115258
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| A115258 |
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Isolated primes in Ulam ' s lattice (1,2,... in spiral). |
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+0
3
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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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Isolated prime numbers have not 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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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.
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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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CROSSREFS
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Cf. A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988, A014848.
Cf. A113688 Isolated semiprimes in the semiprime spiral.
Adjacent sequences: A115255 A115256 A115257 this_sequence A115259 A115260 A115261
Sequence in context: A062677 A045714 A090156 this_sequence A139965 A139765 A031412
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KEYWORD
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nonn
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AUTHOR
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Giorgio Balzarotti and Paolo P. Lava (greenblue(AT)tiscali.it), Feb 17 2006
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