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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 13 19:36 EDT 2021. Contains 343868 sequences. (Running on oeis4.)