

A164119


Numbers k that are the smallest number that produces the ordered pair (d(k), d(k+1)), where d(k) is the number of divisors of k.


3



1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 20, 23, 24, 27, 30, 35, 36, 39, 44, 47, 48, 49, 54, 59, 60, 63, 64, 80, 81, 84, 95, 99, 104, 111, 112, 119, 120, 143, 144, 152, 153, 167, 169, 175, 176, 179, 180, 191, 192, 195, 210, 216, 224, 225, 239, 240, 252, 260, 272, 275
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OFFSET

1,2


COMMENTS

The set of numbers that produces a given ordered pair (i,j) is either empty, finite, or infinite. The pair (3,3) is produced by no number because d(k)=3 only if k is the square of a prime and no two consecutive numbers are squares of primes. Sequence A161460 lists numbers k that produce a unique ordered pair. It appears that the ordered pair (4,2) is produced by an infinite number of k, which is another way of conjecturing that there are an infinite number of safe primes, A005385. The pair (2,4) is produced by primes in A005383. The numbers in A039832 produce the pair (4,4).


LINKS

T. D. Noe, Terms less than 10^8


EXAMPLE

7 is not here because (d(7), d(8)) = (2,4), which is the same ordered pair produced by k=5.


MATHEMATICA

s={}; Reap[Do[pr=DivisorSigma[0, {n, n+1}]; If[ !MemberQ[s, pr], AppendTo[s, pr]; Sow[n]], {n, 1000}]][[2, 1]]


CROSSREFS

Cf. A005385, A039832, A161460.
Sequence in context: A052047 A078831 A031177 * A308017 A184107 A326583
Adjacent sequences: A164116 A164117 A164118 * A164120 A164121 A164122


KEYWORD

nonn


AUTHOR

T. D. Noe, Aug 10 2009


STATUS

approved



