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A137721 Number of numbers not greater than n with no prime gaps in their factorization.

%I A137721 #10 Dec 09 2021 16:15:17
%S A137721 1,2,3,4,5,6,7,8,9,9,10,11,12,12,13,14,15,16,17,17,17,17,18,19,20,20,
%T A137721 21,21,22,23,24,25,25,25,26,27,28,28,28,28,29,29,30,30,31,31,32,33,34,
%U A137721 34,34,34,35,36,36,36,36,36,37,38,39,39,39,40,40,40,41,41,41,41,42,43,44
%N A137721 Number of numbers not greater than n with no prime gaps in their factorization.
%C A137721 a(n) > a(n-1) iff A073490(n) = 0;
%C A137721 a(n) > A137722(n) for n < 134;
%C A137721 a(n) < A137722(n) for n > 140;
%C A137721 a(A137723(n) + n) = a(A137723(n)) + 1.
%C A137721 Partial sums of A137794. - _Reinhard Zumkeller_, Feb 11 2008
%H A137721 Reinhard Zumkeller, <a href="/A137721/b137721.txt">Table of n, a(n) for n = 1..1000</a>
%F A137721 a(n) = Sum_{k=1..n} 0^A073490(k).
%t A137721 b[n_] := With[{pp = PrimePi @ FactorInteger[ n ][[All, 1]]},
%t A137721      Boole[pp[[-1]] - pp[[1]] + 1 == Length[pp]]]; (* b is A137794 *)
%t A137721 Array[b, 105] // Accumulate (* _Jean-François Alcover_, Dec 09 2021 *)
%Y A137721 Cf. A073490, A073491, A137722, A137723, A137794.
%K A137721 nonn
%O A137721 1,2
%A A137721 _Reinhard Zumkeller_, Feb 09 2008