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A331186 Exponent of the highest power of prime(A067004(n)) which divides n, where A067004 is the ordinal transform of number of divisors of n (A000005).

%I A331186 #11 Dec 21 2021 10:15:36
%S A331186 0,1,1,2,1,1,1,0,2,1,1,2,1,1,0,4,1,2,1,1,0,0,1,3,2,0,0,1,1,1,1,0,0,0,
%T A331186 0,2,1,0,0,1,1,1,1,0,0,0,1,4,2,0,0,0,1,0,0,0,0,0,1,2,1,0,0,6,0,0,1,0,
%U A331186 0,0,1,2,1,0,0,0,0,0,1,0,4,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0
%N A331186 Exponent of the highest power of prime(A067004(n)) which divides n, where A067004 is the ordinal transform of number of divisors of n (A000005).
%H A331186 Antti Karttunen, <a href="/A331186/b331186.txt">Table of n, a(n) for n = 1..65537</a>
%F A331186 a(n) = A286561(n, A000040(A067004(n))), where A286561(n,k) gives the k-valuation of n.
%t A331186 b[_] = 0;
%t A331186 A067004[n_] := A067004[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
%t A331186 a[n_] := IntegerExponent[n, Prime[A067004[n]]];
%t A331186 Array[a, 105] (* _Jean-François Alcover_, Dec 21 2021 *)
%o A331186 (PARI)
%o A331186 up_to = 65537;
%o A331186 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A331186 v067004 = ordinal_transform(vector(up_to,n,numdiv(n)));
%o A331186 A067004(n) = v067004[n];
%o A331186 A331186(n) = valuation(n,prime(A067004(n)));
%Y A331186 Cf. A000005, A067004, A286561.
%Y A331186 Cf. also A331185, A331187.
%K A331186 nonn
%O A331186 1,4
%A A331186 _Antti Karttunen_, Jan 12 2020