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A080611
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a(n) is the smallest number m >= 2 for which the set of prime factors of m, m-1 and m+1 contains at least the first n primes.
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0
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2, 2, 4, 6, 21, 155, 441, 2925, 10165, 342056, 2781505, 10631544, 163886800, 498936010, 5163068911, 794010643700, 17635639237580, 353823355745574, 16828233620277430, 224220167903546529, 11990471619719586785, 113367767003198032480, 4446177962278202834685, 118332081735203144063619, 1103720538399012083835935, 78239926422758111576984420
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OFFSET
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1,1
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COMMENTS
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This sequence is of use in non-decimal systems whereby digits in base a(n) can be tested using simple addition tricks [and no higher operations] to determine if the number represented is relatively prime with respect to the first n primes.
The addition trick for base a(n) is to sum digits to do a(n)-1 divisibility tests and alternately add and subtract digits to perform the a(n)+1 test. In base 10 we add digits to find 9-divisibility or add-subtract digits (e.g. 132 = 2-3+1 = 0 is divisible by 11) to find divisibility by 11.
a(5) = 21 because 20, 21 and 22 have between them all 5 prime factors 2,3,5,7,11. - Don Reble, Feb 27 2003
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LINKS
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FORMULA
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a(n) is the smallest number such that the product [a(n)-1]a(n)[a(n)+1] has prime factors which include the first n ordinal primes excluding 1 (see A033946).
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EXAMPLE
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a(1) = 1 since we assume 0 and 1 have no nontrivial prime factors, thus a(1)+1 is the only term with factors, {2}.
a(4) = 6 because a(4)-1 = 5, thus the set of prime factors {5}; a(4) = 2*3, thus the set of prime factors {2, 3} and a(4)+1 = 7 with the set of prime factors {7}. The combined set, {2, 3, 5, 7} contains the first 4 prime numbers (not including 1) and because there are no numbers less than 6 with this property, a(4) = 6.
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PROG
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a = 1 firstNPrimes = getFirstNPrimes(n) /* A033946[1:n+1] */ a = 0 do { ++a; primes = prime_factors(a) primes.union(prime_factors(a-1)) primes.union(prime_factors(a+1)) } while (!(firstNPrimes is_subset_of primes)) return a;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Jeffrey C. Jacobs (darklord(AT)timehorse.com), Feb 26 2003
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EXTENSIONS
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STATUS
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approved
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