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A333235 a(n) is the product of indices of unitary prime power divisors of n.

%I A333235 #15 Sep 06 2024 14:09:01
%S A333235 1,1,2,3,4,2,5,6,7,4,8,6,9,5,8,10,11,7,12,12,10,8,13,12,14,9,15,15,16,
%T A333235 8,17,18,16,11,20,21,19,12,18,24,20,10,21,24,28,13,22,20,23,14,22,27,
%U A333235 24,15,32,30,24,16,25,24,26,17,35,27,36,16,28,33,26,20
%N A333235 a(n) is the product of indices of unitary prime power divisors of n.
%C A333235 Equivalently: replace each prime power p^e in the prime factorization of n by its index in A246655. - _M. F. Hasler_, Jun 16 2021
%H A333235 Robert Israel, <a href="/A333235/b333235.txt">Table of n, a(n) for n = 1..10000</a>
%F A333235 If n = Product (p_j^k_j) then a(n) = Product (A025528(p_j^k_j)).
%F A333235 a(prime(n)) = A027883(n).
%F A333235 a(2^n) = A182908(n).
%F A333235 a(A246655(n)) = n.
%e A333235 a(600) = a(2^3 * 3 * 5^2) = a(A246655(6) * A246655(2) * A246655(14)) = 6 * 2 * 14 = 168.
%p A333235 N:= 1000: # for a(1)..a(N)
%p A333235 R:= NULL: p:= 2:
%p A333235 while p < N do
%p A333235   R:= R,  seq(p^k,k=1..ilog[p](N));
%p A333235   p:= nextprime(p);
%p A333235 od:
%p A333235 L:= sort([R]):
%p A333235 f:= proc(n) local F, t;
%p A333235   F:= ifactors(n)[2];
%p A333235   mul(ListTools:-BinarySearch(L,t[1]^t[2]),t=F)
%p A333235 end proc:
%p A333235 map(f, [$1..N]); # _Robert Israel_, Feb 11 2021
%t A333235 PrimePowerPi[n_] := Sum[Boole[PrimePowerQ[k]], {k, 1, n}]; a[1] = 1; a[n_] := Times @@ (PrimePowerPi[#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]
%o A333235 (PARI) apply( {A333235(n)=vecprod([A322981(f[1]^f[2])|f<-factor(n)~])}, [1..99]) \\ _M. F. Hasler_, Jun 16 2021
%Y A333235 Cf. A003963, A025528, A027883, A141128, A141809, A156061, A182908, A246655.
%Y A333235 Cf. A322981 (the index of n = p^e in A246655).
%K A333235 nonn,mult,look
%O A333235 1,3
%A A333235 _Ilya Gutkovskiy_, Mar 12 2020