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Numbers n such that d(n+1) < 2d(n), where d() is the number of divisors function A000005.
2

%I #13 Aug 19 2020 16:46:07

%S 2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,36,

%T 38,40,42,44,45,46,48,50,51,52,54,56,57,58,60,62,63,64,66,68,70,72,74,

%U 75,76,78,80,81,82,84,85,86,88,90,91,92,93,94,96,98,99,100,102,104,105

%N Numbers n such that d(n+1) < 2d(n), where d() is the number of divisors function A000005.

%H Harry J. Smith, <a href="/A063450/b063450.txt">Table of n, a(n) for n = 1..1000</a>

%e d(p+1) < 2d(p) holds mainly for composites and the primes 2 and 3. E.g.:

%e n=10: 2d(10) = 2*4 = 8 > 2 = d(11);

%e n=3: 2d(3) = 2*2 = 4 > d(4) = 3;

%e n=2: 2d(2) = 2*2 = 4 > d(3) = 2.

%t SequencePosition[DivisorSigma[0,Range[110]],_?(#[[2]]<2#[[1]]&)][[All,1]]// Quiet (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 19 2020 *)

%o (PARI) { n=0; for (m=1, 10^9, if (numdiv(m + 1) < 2*numdiv(m), write("b063450.txt", n++, " ", m); if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 21 2009

%Y Cf. A000005, A002808.

%K nonn

%O 1,1

%A _Labos Elemer_, Jul 24 2001

%E Formatting by _Charles R Greathouse IV_, Mar 24 2010