%I #31 Sep 15 2024 22:00:41

%S 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 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 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 Characteristic function for powerful numbers.

%C 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

%C a(n) * A008966(n) = A063524(n). - _Reinhard Zumkeller_, Sep 16 2011

%H Reinhard Zumkeller, <a href="/A112526/b112526.txt">Table of n, a(n) for n = 1..10000</a>

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number.</a>

%F Multiplicative with a(p^e) = 1 - 0^(e-1), e>0 and p prime.

%F Dirichlet g.f.: zeta(2*s)*zeta(3*s)/zeta(6*s), e.g. A082695 at s=1.

%F a(n) = {m: Min{A124010(m,k): k=1..A001221(m)} > 1}. - _Reinhard Zumkeller_, Jun 03 2015

%F 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

%e a(72)=1 because 72=2^3*3^2 has all exponents > 1.

%t cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1,1,0]; Array[ cfpn,120] (* _Harvey P. Dale_, Jul 17 2012 *)

%o (Haskell)

%o a112526 1 = 1

%o a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n

%o -- _Reinhard Zumkeller_, Jun 03 2015, Sep 16 2011

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1+X^3)/(1-X^2))[n], ", ")) \\ _Vaclav Kotesovec_, Jul 15 2022

%o (Python)

%o from sympy import factorint

%o def A112526(n): return int(all(e>1 for e in factorint(n).values())) # _Chai Wah Wu_, Sep 15 2024

%Y Powerful numbers: A001694. Differs from perfect powers A075802 at Achilles numbers A052486.

%Y Cf. A124010, A001221, A027746.

%Y Cf. A008683, A008966, A005596, A065471, A082695, A063524.

%K mult,nonn

%O 1,1

%A _Franklin T. Adams-Watters_, Sep 09 2005