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A112526 Characteristic function for powerful numbers.

%I A112526 #47 Jul 03 2025 09:29:52
%S A112526 1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,
%T A112526 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%U A112526 0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0
%N A112526 Characteristic function for powerful numbers.
%C A112526 A signed multiplicative variant is defined by b(n) = a(n)*mu(n) with mu = A008683, such that b(p^e)=0 if e=1 and b(p^e)= -1 if e>1. This has Dirichlet series Sum_{n>=1} b(n)/n = A005596 and Sum_{n>=1} b(n)/n^2 = A065471. - _R. J. Mathar_, Apr 04 2011
%H A112526 Reinhard Zumkeller, <a href="/A112526/b112526.txt">Table of n, a(n) for n = 1..10000</a>
%H A112526 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number</a>.
%H A112526 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A112526 Multiplicative with a(p^e) = 1 - 0^(e-1), e > 0 and p prime.
%F A112526 Dirichlet g.f.: zeta(2*s)*zeta(3*s)/zeta(6*s), e.g., A082695 at s=1.
%F A112526 a(n) * A008966(n) = A063524(n). - _Reinhard Zumkeller_, Sep 16 2011
%F A112526 a(n) = {m: Min{A124010(m,k): k=1..A001221(m)} > 1}. - _Reinhard Zumkeller_, Jun 03 2015
%F A112526 Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)/zeta(3) + 6*zeta(2/3)*n^(1/3)/Pi^2. - _Vaclav Kotesovec_, Feb 08 2019
%F A112526 a(n) = Sum_{d|n} A005361(d)*A008683(n/d). - _Ridouane Oudra_, Jul 03 2025
%e A112526 a(72) = 1 because 72 = 2^3*3^2 has all exponents > 1.
%t A112526 cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1,1,0]; Array[ cfpn,120] (* _Harvey P. Dale_, Jul 17 2012 *)
%o A112526 (Haskell)
%o A112526 a112526 1 = 1
%o A112526 a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n
%o A112526 -- _Reinhard Zumkeller_, Jun 03 2015, Sep 16 2011
%o A112526 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1+X^3)/(1-X^2))[n], ", ")) \\ _Vaclav Kotesovec_, Jul 15 2022
%o A112526 (PARI) a(n) = ispowerful(n); \\ _Amiram Eldar_, Jul 02 2025
%o A112526 (Python)
%o A112526 from sympy import factorint
%o A112526 def A112526(n): return int(all(e>1 for e in factorint(n).values())) # _Chai Wah Wu_, Sep 15 2024
%Y A112526 Differs from characteristic function of perfect powers A075802 at Achilles numbers A052486.
%Y A112526 Cf. A001694 (powerful numbers), A124010, A001221, A027746.
%Y A112526 Cf. A008683, A008966, A005596, A065471, A082695, A063524.
%Y A112526 Cf. A002117, A078434, A005361.
%K A112526 mult,nonn,easy
%O A112526 1,1
%A A112526 _Franklin T. Adams-Watters_, Sep 09 2005