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%I #5 Oct 19 2017 03:14:12
%S 2,2,4,6,21,155,441,2925,10165,342056,2781505,10631544,163886800,
%T 498936010,5163068911,794010643700,17635639237580,353823355745574,
%U 16828233620277430,224220167903546529,11990471619719586785,113367767003198032480,4446177962278202834685,118332081735203144063619,1103720538399012083835935,78239926422758111576984420
%N 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.
%C 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.
%C 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.
%C 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
%H Jeffrey C. Jacobs, <a href="http://www.timehorse.com/">Time Horse Home</a>.
%H Robert Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Interesting Numbers</a>.
%F 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).
%e 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}.
%e 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.
%o 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;
%Y Cf. A033946.
%K nonn
%O 1,1
%A Jeffrey C. Jacobs (darklord(AT)timehorse.com), Feb 26 2003
%E More terms from _Don Reble_, Feb 27 2003
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