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

2,2

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

Cf. A051903, A334572, A334574.

Sequence in context: A345918 A269389 A092054 * A228370 A186112 A029453

Adjacent sequences: A334570 A334571 A334572 * A334574 A334575 A334576

Keyword

nonn

Author

Michel Marcus, May 06 2020

Status

approved