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%I #67 Mar 31 2023 14:14:35
%S 1,1,1,1,1,0,1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,
%T 0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,1,0,
%U 0,0,1,0,1,0,0,0,0,0,1,0
%N 1 if n is a prime power p^k (k >= 0), otherwise 0.
%C Characteristic function of unit or prime powers p^k (k >= 1). Characteristic function of prime powers p^k (k >= 0). - _Daniel Forgues_, Mar 03 2009
%C See A065515 for partial sums. - _Reinhard Zumkeller_, Nov 22 2009
%H Reinhard Zumkeller, <a href="/A010055/b010055.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F Dirichlet generating function: 1 + ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s-1) = Sum_{k>=1} primezeta(k*s). - _Franklin T. Adams-Watters_, Sep 11 2005
%F a(n) = 0^(A119288(n)-1). - _Reinhard Zumkeller_, May 13 2006
%F a(A000961(n)) = 1; a(A024619(n)) = 0. - _Reinhard Zumkeller_, Nov 17 2011
%F a(n) = if A001221(n) <= 1 then 1, otherwise 0. - _Reinhard Zumkeller_, Nov 28 2015
%F If n >= 2, a(n) = A069513(n). - _Jeppe Stig Nielsen_, Feb 02 2016
%F Conjecture: a(n) = (n - A048671(n))/A000010(n) for all n > 1. - _Velin Yanev_, Mar 10 2021 [The conjecture is true. - _Andrey Zabolotskiy_, Mar 11 2021]
%p A010055 := proc(n)
%p if n =1 then
%p 1;
%p else
%p if nops(ifactors(n)[2]) = 1 then
%p 1;
%p else
%p 0 ;
%p end if;
%p end if;
%p end proc: # _R. J. Mathar_, May 25 2017
%t A010055[n_]:=Boole[PrimeNu[n]<=1]; A010055/@Range[20] (* _Enrique Pérez Herrero_, May 30 2011 *)
%t {1}~Join~Table[Boole@ PrimePowerQ@ n, {n, 2, 105}] (* _Michael De Vlieger_, Feb 02 2016 *)
%o (PARI) for(n=1,120,print1(omega(n)<=1,","))
%o (Haskell)
%o a010055 n = if a001221 n <= 1 then 1 else 0
%o -- _Reinhard Zumkeller_, Nov 28 2015, Mar 19 2013, Nov 17 2011
%o (Python)
%o from sympy import primefactors
%o def A010055(n): return int(len(primefactors(n)) <= 1) # _Chai Wah Wu_, Mar 31 2023
%Y Cf. A069513 (1 if n is a prime power p^k (k >= 1), else 0.)
%Y Cf. A268340.
%Y Cf. A100995.
%Y Cf. A001221, A000961, A024619, A065515,
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Charles R Greathouse IV_, Mar 12 2008
%E Edited by _Daniel Forgues_, Mar 02 2009
%E Comment re Galois fields moved to A069513 by _Franklin T. Adams-Watters_, Nov 02 2009
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