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Characteristic function of numbers having no prime gaps in their factorization.
7

%I #13 Dec 09 2021 11:54:33

%S 1,1,1,1,1,1,1,1,1,0,1,1,1,0,1,1,1,1,1,0,0,0,1,1,1,0,1,0,1,1,1,1,0,0,

%T 1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0,0,1,1,1,0,0,1,0,0,1,0,

%U 0,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,1,0,1,0,1

%N Characteristic function of numbers having no prime gaps in their factorization.

%H Antti Karttunen, <a href="/A137794/b137794.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(n) = 0^A073490(n).

%F a(A073491(n)) = 1; a(A073492(n)) = 0;

%F a(n) = A137721(n) - A137721(n-1) for n>1.

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

%t Boole[pp[[-1]] - pp[[1]] + 1 == Length[pp]]];

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

%o (PARI) A137794(n) = if(1>=omega(n),1,my(pis=apply(primepi,factor(n)[,1])); for(k=2,#pis,if(pis[k]>(1+pis[k-1]),return(0))); (1)); \\ _Antti Karttunen_, Sep 27 2018

%Y Cf. A137721 (partial sums).

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Feb 11 2008