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A189827
a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.
5
3, 5, 4, 7, 4, 8, 5, 8, 5, 10, 4, 10, 6, 9, 6, 11, 4, 12, 6, 10, 6, 12, 5, 12, 7, 10, 6, 14, 4, 14, 6, 10, 8, 13, 6, 13, 6, 12, 6, 16, 4, 14, 8, 10, 8, 14, 5, 16, 7, 12, 6, 14, 6, 16, 8, 12, 6, 16, 4, 16, 8, 11, 10, 15, 6, 14, 6, 14, 6, 20, 4, 16, 8, 10, 10
OFFSET
2,1
COMMENTS
d(n-1) + d(n+1) is a measure of the compositeness of the numbers next to n. Sequence A189825 lists the first occurrence of each number.
It is conjectured that every number greater than 3 occurs an infinite number of times. Note that an infinite number of 4's is equivalent to there being an infinite number of twin primes (A001097). An infinite number of 5's is equivalent to there being an infinite number of primes of the form p^2-2 (A028871) or p^2+2 (A056899) for prime p. An infinite number of 6's is equivalent to there being an infinite number of primes of the form p^3-2 (A066878), p^3+2 (A048636), p*q-2 (A063637), or p*q+2 (A063638), where p and q are distinct primes.
EXAMPLE
a(5) = d(4) + d(6) = 3 + 4 = 7.
MATHEMATICA
Table[DivisorSigma[0, n-1] + DivisorSigma[0, n+1], {n, 2, 100}]
First[#]+Last[#]&/@Partition[DivisorSigma[0, Range[80]], 3, 1] (* Harvey P. Dale, May 27 2013 *)
CROSSREFS
Sequence in context: A114748 A182733 A107431 * A023859 A096457 A277897
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 28 2011
STATUS
approved