OFFSET
1,4
COMMENTS
For n>1, such a k_i always exists for every p_i|n, since with k_i=p_i - 1, p-k_i =1, always divides n - p_i. omega(n)<=a(n)<=Sopf(n) - Omega(n). The left side equality applies when n is a prime or a Carmichael number. The right side equality applies to numbers n such that k_i = p_i - 1, 1 <= i <= r (it can be shown that all numbers with this property are even, see A309239). For n>2, records of a(n) occur when n is an even semiprime.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
EXAMPLE
For n prime a(n) = 1; n = 4 = 2*2 —> k_1 = k_2 = 1, so a(4) = 1 + 1 = 2.
MATHEMATICA
g[n_, p_] := Module[{k=1}, While[!Divisible[n - k, p - k], k++]; k]; a[1]=0; a[n_] := Module[{f = FactorInteger[n]}, p=f[[;; , 1]]; e=f[[;; , 2]]; Sum[e[[k]] * g[n, p[[k]]], {k, 1, Length[p]}]]; Array[a, 85] (* Amiram Eldar, Jul 18 2019 *)
PROG
(PARI) getk(p, n) = {my(k=1); while ((n - k) % (p - k), k++); k; }
a(n) = {my(f=factor(n)); for (i=1, #f~, f[i, 1] = getk(f[i, 1], n); ); sum(i=1, #f~, f[i, 1]*f[i, 2]); } \\ Michel Marcus, Jul 16 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Jul 14 2019
EXTENSIONS
More terms from Michel Marcus, Jul 16 2019
STATUS
approved