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%I #13 Feb 09 2021 11:02:38
%S 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,
%T 54,59,60,63,64,80,81,84,95,99,104,111,112,119,120,143,144,152,153,
%U 167,169,175,176,179,180,191,192,195,210,216,224,225,239,240,252,260,272,275
%N 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.
%C 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).
%H T. D. Noe, <a href="/A164119/b164119.txt">Terms less than 10^8</a>
%e 7 is not here because (d(7), d(8)) = (2,4), which is the same ordered pair produced by k=5.
%t s={}; Reap[Do[pr=DivisorSigma[0,{n,n+1}]; If[ !MemberQ[s,pr], AppendTo[s,pr]; Sow[n]], {n,1000}]][[2,1]]
%Y Cf. A005385, A039832, A161460.
%K nonn
%O 1,2
%A _T. D. Noe_, Aug 10 2009
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