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Search: id:A086320
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| A086320 |
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a(n) = depth of the prime tree formed when 4p+/-3 is applied to the n-th prime, and repeatedly to any primes generated from the n-th prime via this process. |
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+0
1
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| 11, 1, 6, 3, 10, 1, 3, 6, 5, 3, 2, 3, 2, 1, 9, 1, 6, 3, 3, 2, 1, 5, 1, 4, 1, 3, 2, 3, 4, 2, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 5, 2, 8, 3, 1, 1, 1, 1, 2, 3, 5, 2, 2, 1, 3, 2, 1, 2, 1, 1, 4, 1, 2, 1, 4, 1, 5, 1, 1, 2, 3, 2, 3, 3, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 4, 2, 1, 3, 1, 2, 2, 6, 4, 1, 1, 1, 2
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Note all prime trees have a minimum depth of 1, as the starting prime forms the root of the tree. Note also that of the 125 primes tested, 77 have trees greater than 1 generation in depth. This implies one's odds of generating a prime number by applying 4p+/-3 to a prime are slightly better than even at 61.6%. It would be interesting to test this to a larger number of primes and see if the percentage rises or falls.
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LINKS
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C. Seggelin, First 125 Prime Trees Formed by 4p+/-3.
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EXAMPLE
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a(125) = 5 because the 125-th prime is 691 which generates further primes through 4 repeated applications of 4p+/-3, giving a prime tree with generations as follows:
1. 691
2. 4 x 691 + 3 = 2767
3. 4 x 2767 + 3 = 11071
4. 4 x 11071 - 3 = 44281
5. 4 x 44281 + 3 = 177127
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MAPLE
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// PSEUDOCODE BEGIN getPrimeTreeDepth(coeff, offset, root): // recursive let result=0; if (isprime(root)) then let minusBDepth = getPrimeTreeDepth(coeff, offset, (coeff * root - offset)); let plusBDepth = getPrimeTreeDepth(coeff, offset, (coeff * root + offset)); result = MAX(minusBDepth, plusBDepth) + 1; // add one to count this node end if; return result; END. let c=4; let d=3; for n=1 to 125 let p=nthPrime(n); depth=getPrimeTreeDepth(c, d, p); sequence.add(depth); next n.
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CROSSREFS
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Cf. A086319.
Sequence in context: A010199 A113492 A010200 this_sequence A095193 A010201 A063431
Adjacent sequences: A086317 A086318 A086319 this_sequence A086321 A086322 A086323
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KEYWORD
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nonn
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AUTHOR
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Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 17 2003
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