cp's OEIS Frontend

A069513 Characteristic function of the prime powers p^k, k >= 1.

%I A069513 #72 Aug 05 2025 18:58:55
%S A069513 0,1,1,1,1,0,1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,
%T A069513 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 A069513 0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0
%N A069513 Characteristic function of the prime powers p^k, k >= 1.
%C A069513 Also, number of Galois fields of order n. - _Charles R Greathouse IV_, Mar 12 2008
%C A069513 Also, number of abelian indecomposable groups of order n. - _Kevin Lamoreau_, Mar 13 2023
%H A069513 Daniel Forgues, <a href="/A069513/b069513.txt">Table of n, a(n) for n = 1..100000</a>.
%F A069513 If n >= 2, a(n) = A010055(n).
%F A069513 a(n) = Sum_{d|n} bigomega(d)*mu(n/d); equivalently, Sum_{d|n} a(d) = bigomega(n); equivalently, Möbius transform of bigomega(n).
%F A069513 Dirichlet g.f.: 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 A069513 a(n) = floor(1/A001221(n)), for n > 1. - _Enrique Pérez Herrero_, Jun 01 2011
%F A069513 a(n) = - Sum_{d|n} mu(d)*bigomega(d), where bigomega = A001222. - _Ridouane Oudra_, Oct 29 2024
%F A069513 a(n) = - Sum_{d|n} mu(d)*omega(d), where omega = A001221. - _Ridouane Oudra_, Jul 30 2025
%p A069513 A069513 := proc(n)
%p A069513     if n = 1 then
%p A069513         0 ;
%p A069513     elif A001221(n) > 1 then
%p A069513         0;
%p A069513     else
%p A069513         1 ;
%p A069513     end if ;
%p A069513 end proc:
%p A069513 seq(A069513(n),n=1..80) ; # _R. J. Mathar_, Nov 02 2016
%t A069513 A069513[n_]:=Boole[PrimeNu[n]==1]; A069513/@Range[20] (* _Enrique Pérez Herrero_, May 30 2011 *)
%o A069513 (PARI) for(n=1,120,print1(omega(n)==1,","))
%o A069513 (Haskell)
%o A069513 a069513 1 = 0
%o A069513 a069513 n = a010055 n  -- _Reinhard Zumkeller_, Mar 19 2013
%o A069513 (Python)
%o A069513 from sympy import primefactors
%o A069513 def A069513(n): return int(len(primefactors(n)) == 1) # _Chai Wah Wu_, Mar 31 2023
%Y A069513 Cf. A246655, A010055, A001221, A001222, A008683, A090751, A000688, A361414.
%Y A069513 The partial sums of this sequence give A025528. - _Daniel Forgues_, Mar 02 2009
%Y A069513 Cf. A014963, A003418. - _Enrique Pérez Herrero_, Jun 01 2011
%K A069513 easy,nonn
%O A069513 1,1
%A A069513 _Benoit Cloitre_, Apr 15 2002
%E A069513 Moved original definition to formula line. Used comment (that I previously added) as definition. - _Daniel Forgues_, Mar 08 2009
%E A069513 Edited by _Franklin T. Adams-Watters_, Nov 02 2009