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Number of numbers not greater than n with no prime gaps in their factorization.
5

%I #10 Dec 09 2021 16:15:17

%S 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 21,21,22,23,24,25,25,25,26,27,28,28,28,28,29,29,30,30,31,31,32,33,34,

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

%C a(n) > a(n-1) iff A073490(n) = 0;

%C a(n) > A137722(n) for n < 134;

%C a(n) < A137722(n) for n > 140;

%C a(A137723(n) + n) = a(A137723(n)) + 1.

%C Partial sums of A137794. - _Reinhard Zumkeller_, Feb 11 2008

%H Reinhard Zumkeller, <a href="/A137721/b137721.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{k=1..n} 0^A073490(k).

%t b[n_] := With[{pp = PrimePi @ FactorInteger[ n ][[All, 1]]},

%t Boole[pp[[-1]] - pp[[1]] + 1 == Length[pp]]]; (* b is A137794 *)

%t Array[b, 105] // Accumulate (* _Jean-François Alcover_, Dec 09 2021 *)

%Y Cf. A073490, A073491, A137722, A137723, A137794.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Feb 09 2008