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A317833 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078898 (the ordinal transform of A020639, the smallest prime factor of n).

%I A317833 #15 Dec 19 2021 04:34:23
%S A317833 1,1,1,7,1,5,1,25,7,9,1,31,1,13,5,363,1,55,1,55,7,21,1,101,7,25,33,79,
%T A317833 1,41,1,1335,11,33,5,305,1,37,13,177,1,59,1,127,47,45,1,1371,7,175,17,
%U A317833 151,1,309,7,253,19,57,1,187,1,61,67,9923,9,95,1,199,23,113,1,927,1,73,87,223,5,113,1,2379,715,81,1,265,11
%N A317833 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078898 (the ordinal transform of A020639, the smallest prime factor of n).
%C A317833 The first negative term is a(840) = -445.
%H A317833 Antti Karttunen, <a href="/A317833/b317833.txt">Table of n, a(n) for n = 1..16384</a>
%F A317833 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078898(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%t A317833 lpf[n_] := If[n == 1, 1, FactorInteger[n][[1, 1]]];
%t A317833 b[_] = 1;
%t A317833 A078898[n_] := A078898[n] = If[n == 0, 0, With[{t = lpf[n]}, b[t]++]];
%t A317833 f[n_] := f[n] = If[n == 1, 1, (1/2)(A078898[n] - Sum[If[1 < d < n, f[d]*f[n/d], 0], {d, Divisors[n]}])]
%t A317833 a[n_] := Numerator[f[n]];
%t A317833 Array[a, 100] (* _Jean-François Alcover_, Dec 19 2021 *)
%o A317833 (PARI)
%o A317833 up_to = 16384;
%o A317833 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 A317833 A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
%o A317833 v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A317833 A078898(n) = v078898[n];
%o A317833 A317833aux(n) = if(1==n,n,(A078898(n)-sumdiv(n,d,if((d>1)&&(d<n),A317833aux(d)*A317833aux(n/d),0)))/2);
%o A317833 A317833(n) = numerator(A317833aux(n));
%Y A317833 Cf. A046644, A078898.
%Y A317833 Cf. also A305798, A305803, A305804, A317830, A317834.
%K A317833 sign,frac
%O A317833 1,4
%A A317833 _Antti Karttunen_, Aug 10 2018