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A334573 Partial sums of A334572.

%I A334573 #19 Jan 05 2024 02:52:17
%S A334573 1,2,4,6,7,8,11,14,16,17,19,21,22,23,27,31,33,35,37,39,40,41,44,47,49,
%T A334573 52,55,57,58,59,64,69,70,71,73,75,76,77,80,83,84,85,87,89,91,92,96,
%U A334573 100,102,104,106,108,111,114,117,120,121,122,124,126,127,129,135,141
%N A334573 Partial sums of A334572.
%C A334573 a(n) = L_infinite(n) = Sum_{m=2..n} d_infinite(m, m-1) as defined in Kolossváry link.
%H A334573 Charles R Greathouse IV, <a href="/A334573/b334573.txt">Table of n, a(n) for n = 2..10000</a>
%H A334573 István B. Kolossváry and István T. Kolossváry, <a href="https://doi.org/10.1016/j.jnt.2021.09.010">Distance between natural numbers based on their prime signature</a>, Journal of Number Theory, Vol. 234 (2022), pp. 120-139; <a href="https://arxiv.org/abs/2005.02027">arXiv preprint</a>, arXiv:2005.02027 [math.NT], 2020-2021.
%F A334573 a(n) = Sum_{m=2..n} A334572(n).
%F A334573 a(n) = Sum_{m=2..n} max(A051903(n), A051903(n-1)).
%F A334573 a(n) ~ c * n, where c = 2.2883695... (A334574). - _Amiram Eldar_, Jan 05 2024
%p A334573 f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
%p A334573 b:= n-> max(map(abs, {coeffs(f(n)-f(n-1))})):
%p A334573 a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+b(n)) end:
%p A334573 seq(a(n), n=2..80);  # _Alois P. Heinz_, May 06 2020
%t A334573 f[n_] := Sum[{p, e} = pe; e x^p, {pe, FactorInteger[n]}];
%t A334573 b[n_] :=  CoefficientList[f[n] - f[n-1], x] // Abs // Max;
%t A334573 b /@ Range[2, 80] // Accumulate (* _Jean-François Alcover_, Nov 16 2020, after _Alois P. Heinz_ *)
%t A334573 Accumulate[Max @@@ Partition[Join[{0}, Table[Max[FactorInteger[n][[;; , 2]]], {n, 2, 100}]], 2, 1]] (* _Amiram Eldar_, Jan 05 2024 *)
%o A334573 (PARI) d(n) = {my(f=factor(n/(n-1))[,2]~); vecmax(apply(x->abs(x), f));}
%o A334573 a(n) = sum(k=2, n, d(k));
%o A334573 (PARI) first(n)=my(v=vector(n-1),o,t,s); forfactored(k=2,n, t=vecmax(k[2][,2]); v[k[1]-1]=s+=max(o,t); o=t); v \\ _Charles R Greathouse IV_, Feb 01 2022
%Y A334573 Cf. A051903, A334572, A334574.
%K A334573 nonn
%O A334573 2,2
%A A334573 _Michel Marcus_, May 06 2020