login
A334573
Partial sums of A334572.
3
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, 52, 55, 57, 58, 59, 64, 69, 70, 71, 73, 75, 76, 77, 80, 83, 84, 85, 87, 89, 91, 92, 96, 100, 102, 104, 106, 108, 111, 114, 117, 120, 121, 122, 124, 126, 127, 129, 135, 141
OFFSET
2,2
COMMENTS
a(n) = L_infinite(n) = Sum_{m=2..n} d_infinite(m, m-1) as defined in Kolossváry link.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
István B. Kolossváry and István T. Kolossváry, Distance between natural numbers based on their prime signature, Journal of Number Theory, Vol. 234 (2022), pp. 120-139; arXiv preprint, arXiv:2005.02027 [math.NT], 2020-2021.
FORMULA
a(n) = Sum_{m=2..n} A334572(n).
a(n) = Sum_{m=2..n} max(A051903(n), A051903(n-1)).
a(n) ~ c * n, where c = 2.2883695... (A334574). - Amiram Eldar, Jan 05 2024
MAPLE
f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
b:= n-> max(map(abs, {coeffs(f(n)-f(n-1))})):
a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+b(n)) end:
seq(a(n), n=2..80); # Alois P. Heinz, May 06 2020
MATHEMATICA
f[n_] := Sum[{p, e} = pe; e x^p, {pe, FactorInteger[n]}];
b[n_] := CoefficientList[f[n] - f[n-1], x] // Abs // Max;
b /@ Range[2, 80] // Accumulate (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)
Accumulate[Max @@@ Partition[Join[{0}, Table[Max[FactorInteger[n][[;; , 2]]], {n, 2, 100}]], 2, 1]] (* Amiram Eldar, Jan 05 2024 *)
PROG
(PARI) d(n) = {my(f=factor(n/(n-1))[, 2]~); vecmax(apply(x->abs(x), f)); }
a(n) = sum(k=2, n, d(k));
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, May 06 2020
STATUS
approved