A080997 The positive integers arranged in nonincreasing order of centrality (the fraction of n represented by the average gcd of n and the other positive integers).

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 9, 15, 14, 18, 16, 20, 24, 11, 30, 13, 21, 28, 22, 36, 17, 40, 27, 26, 42, 32, 48, 25, 19, 60, 33, 35, 45, 44, 34, 23, 54, 56, 39, 72, 50, 38, 52, 84, 66, 70, 90, 63, 29, 80, 46, 31, 51, 64, 120, 55, 78, 96, 75, 68, 57, 108, 49, 88, 37, 65, 105

Offset

1,2

Comments

Equivalent descriptions of the centrality of n: 1) Probability that a randomly chosen product in the multiplication table for positive integers (A003991; see also A061017) is a multiple of n.

2) Probability taken over all exponential numerical bases that if the last digit of a number represents n, the number is a multiple of n. (For example, in base 10, the probability of a number that ends in 5 being a multiple of 5 is 1. Over all possible bases, the fraction of numbers ending in 5 that are multiples of 5 is the centrality of 5, 9/25 or .36.)

An infinite number of integers have the same centrality as at least one other integer. The only such examples in the first 114 terms of the sequence are 64 and 120, which share a centrality of .0625; they are listed in numerical order.

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Formula

Formula for centrality of n: A018804(n)/n^2, where A018804(n) is the sum of gcd (k, n) for 1 <= k <= n.

The centrality of a(n) is given by A080999(n)/(a(n))^2.

Example

The number 6 has a gcd of 1 with all numbers congruent to 1 or 5 modulo 6, 2 with all numbers congruent to 2 or 4 mod 6, 3 with all 3 mod 6 numbers and 6 with all numbers congruent to 0 mod 6. Its average gcd with other integers is 2.5 (A018804(6)/6), which represents 5/12 or .41666... of 6. This places 6 fifth in centrality among the integers, behind 1 (whose centrality is 1), 2 (.75), 3 (5/9 or .555...) and 4 (.5); it is therefore listed fifth in the sequence.

Mathematica

maxTerms = 100; Clear[c, s]; c[n_] := c[n] = Sum[d*EulerPhi[n/d], {d, Divisors[n] }]/n^2; s[terms_] := s[terms] = Sort[Range[terms], c[#1] >= c[#2] & ][[1 ;; maxTerms]]; s[terms = maxTerms]; s[terms += maxTerms]; While[s[terms] != s[terms - maxTerms], terms += maxTerms]; A080997 = s[terms] (* Jean-François Alcover, Feb 19 2015 *)

Crossrefs

Cf. A018804, A080999 for a formula for the numerator of the unreduced centrality fraction. Other related sequences are A080998, A081000, A081001, A081028, A081029.

Sequence in context: A224531 A058213 A348711 * A151942 A054582 A257797

Adjacent sequences: A080994 A080995 A080996 * A080998 A080999 A081000

Keyword

nice,nonn

Author

Matthew Vandermast, Feb 28 2003

Status

approved