A334573 Partial sums of A334572.

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

Author

Michel Marcus, May 06 2020

Keywords

• nonn

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.

Crossrefs

Cf. A051903, A334572, A334574.

Programs

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 *)

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

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