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Search: id:A086004
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| A086004 |
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3-Rotation Cycle Primes. Primes which when added to their own rotation, yield a prime which when added to its own rotation also yields a prime, which in turn when added to its own rotation also yields a prime. |
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| 12917, 12919, 18911, 18913, 22907, 24907, 26903, 28901, 1088063, 1288043, 1408031, 1428029, 1528019, 100083679, 100280419, 100283849, 100483847, 100692793, 100880413, 101080159, 101283839, 101683093, 101683663, 102080149
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Rotation results in the suffix and prefix swapping positions around a central pivot. If the number of digits in n is odd, the center digit remains untouched during rotation. So for example the rotation of 1234 is 3412, while the rotation of 12345 is 45312. As a final step, n is added to its own rotation, so rotateAndAdd(12345) = 12345 + 45312 = 57657. This final number is tested for primality, and if it is prime, the process may be repeated. Note: Have not yet found any 4-Rotation Cycle Primes. CONJECTURE 1: Rotation and addition of primes with even numbers of digits never yields a prime. CONJECTURE 2: There are no 5-Rotation Cycle Primes.
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FORMULA
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BEGIN rotateAndAdd(n) Let e = floor(LOG10(n))+1; Let h = floor(e/2); Let prefix = (n mod 10^h) * (10^(e-h)); Let middle = n - (floor(n/(10^(e-h)))*(10^(e-h))) - (n mod 10^h); Let suffix = floor(n / (10^(e-h))); Let rotated = prefix + middle + suffix; Return (n + rotated); END. If isPrime(n) And isPrime(rotateAndAdd(n)) And isPrime(rotateAndAdd(rotateAndAdd(n))) And isPrime(rotateAndAdd(rotateAndAdd(rotateAndAdd(n)))) Then sequence.add(n)
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EXAMPLE
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a(1)=12917 because i. 12917 is prime. ii. rotate(12917) = 17912 and 12917+17912=30829 which is also prime. iii. rotate(30829) = 29830 and 30829+29830=60659 which is also prime. iv. rotate(60659) = 59660 and 60659+59660 = 120319 which is also prime.
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CROSSREFS
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Cf. A086002, A086003.
Sequence in context: A068359 A140917 A031805 this_sequence A090887 A028385 A138762
Adjacent sequences: A086001 A086002 A086003 this_sequence A086005 A086006 A086007
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KEYWORD
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base,nonn,uned
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AUTHOR
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Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 07 2003
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