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%I #64 Sep 18 2023 02:03:07
%S 1,1,1,4,1,1,1,8,9,1,1,4,1,1,1,16,1,9,1,4,1,1,1,8,25,1,27,4,1,1,1,32,
%T 1,1,1,36,1,1,1,8,1,1,1,4,9,1,1,16,49,25,1,4,1,27,1,8,1,1,1,4,1,1,9,
%U 64,1,1,1,4,1,1,1,72,1,1,25,4,1,1,1,16,81,1,1,4,1,1,1,8,1,9,1,4,1,1,1,32,1
%N Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=b*c^2*d^3 then a(n)=c^2*d^3 when b is minimized.
%H Antti Karttunen, <a href="/A057521/b057521.txt">Table of n, a(n) for n = 1..16383</a> (first 1000 terms from T. D. Noe)
%H Antti Karttunen, <a href="/A057521/a057521.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H Christian Krause, <a href="https://github.com/ckrause/loda">LODA, an assembly language, a computational model and a tool for mining integer sequences</a>.
%H Victor Ufnarovski and Bo Åhlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seq., Vol. 6 (2003), Article 03.3.4.
%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F a(n) = n / A055231(n).
%F Multiplicative with a(p)=1 and a(p^e)=p^e for e>1. - _Vladeta Jovovic_, Nov 01 2001
%F From _Antti Karttunen_, Nov 22 2017: (Start)
%F a(n) = A064549(A003557(n)).
%F A003557(a(n)) = A003557(n).
%F (End)
%F a(n) = gcd(n, A003415(n)^k), for all k >= 2. [This formula was found in the form k=3 by Christian Krause's LODA miner. See Ufnarovski and Åhlander paper, Theorem 5 on p. 4 for why this holds] - _Antti Karttunen_, Mar 09 2021
%F Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/ p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1)). - _Amiram Eldar_, Sep 18 2023
%e a(40) = 8 since 40 = 2^3 * 5 so the powerful part is 2^3 = 8.
%p A057521 := proc(n)
%p local a,d,e,p;
%p a := 1;
%p for d in ifactors(n)[2] do
%p e := d[1] ;
%p p := d[2] ;
%p if e > 1 then
%p a := a*p^e ;
%p end if;
%p end do:
%p return a;
%p end proc: # _R. J. Mathar_, Jun 09 2016
%t rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := n/Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 97}] (* _Jean-François Alcover_, Jun 20 2013 *)
%t f[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 21 2020 *)
%o (PARI) a(n)=my(f=factor(n));prod(i=1,#f~,if(f[i,2]>1,f[i,1]^f[i,2],1)) \\ _Charles R Greathouse IV_, Aug 13 2013
%o (PARI) a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); factorback(f); \\ _Michel Marcus_, Jan 29 2021
%o (Python)
%o from sympy import factorint, prod
%o def a(n): return 1 if n==1 else prod(1 if e==1 else p**e for p, e in factorint(n).items())
%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Jul 19 2017
%o (Python)
%o from math import prod
%o from sympy import factorint
%o def A057521(n): return n//prod(p for p, e in factorint(n).items() if e == 1) # _Chai Wah Wu_, Nov 14 2022
%Y Cf. A001694, A003415, A003557, A055231, A064549.
%K nonn,mult,easy
%O 1,4
%A _Henry Bottomley_, Sep 01 2000
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