A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

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

Author

T. D. Noe, Apr 28 2011

Keywords

• nonn

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.

Examples

			a(5) = d(4) + d(6) = 3 + 4 = 7.
		

Links

• T. D. Noe, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A175144, A189825.

Programs

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 *)