OFFSET
1,2
COMMENTS
A permutation of sequence A018804, which gives the sum of gcd (k,n) for 1 <= k <= n.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14.
FORMULA
The multiplicative formula for the numerator in a positive integer's centrality fraction is: for prime p, a(p^e) = p^(e-1)*((p-1)e+p) (cf. A018804). Dividing by the square of the integer gives the integer's centrality, which is defined to be the average fraction of the integer that it shares with the other integers as a gcd; see A080997 for other interpretations. This sequence gives the unreduced centrality numerators for A080997(n), where A080997 is the sequence of positive integers in nonincreasing order of their centrality.
MATHEMATICA
nn = 120; f[n_] := f[n] = DivisorSum[n, #*EulerPhi[n/#] &]/n^2; s[x_] := s[x] = Sort[Range[x], f[#1] >= f[#2] &][[;; nn]]; s[n = nn]; s[n += nn]; While[s[n] != s[n - nn], n += nn]; Map[Times @@ MapApply[Function[{p, m}, p^(m - 1)*((p - 1) m + p)], FactorInteger[#]] &, s[n]] (* Michael De Vlieger, Oct 08 2025, after Amiram Eldar at A018804 and Jean-François Alcover at A080997 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Vandermast, Mar 02 2003
STATUS
approved