Search: id:A080997
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%I A080997
%S A080997 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,
%T A080997 27,26,42,32,48,25,19,60,33,35,45,44,34,23,54,56,39,72,50,38,52,84,66,
%U A080997 70,90,63,29,80,46,31,51,64,120,55,78,96,75,68,57,108,49,88,37,65,105
%N 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).
%C A080997 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.
%C A080997 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.)
%C A080997 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.
%H A080997 T. D. Noe, <a href="b080997.txt">Table of n, a(n) for n=1..1000</a>
%F A080997 Formula for centrality of n: A018804(n)/n^2, where A018804(n) is the 
               sum of gcd (k, n) for 1 <= k <= n.
%F A080997 The centrality of a(n) is given by A080999(n)/(a(n))^2.
%e A080997 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.
%Y A080997 Cf. A018804, A080999 for a formula for the numerator of the unreduced 
               centrality fraction. Other related sequences are A080998, A081000, 
               A081001, A081028, A081029.
%Y A080997 Sequence in context: A080738 A032447 A058213 this_sequence A151942 A054582 
               A099884
%Y A080997 Adjacent sequences: A080994 A080995 A080996 this_sequence A080998 A080999 
               A081000
%K A080997 nice,nonn
%O A080997 1,2
%A A080997 Matthew Vandermast (ghodges14(AT)comcast.net), Feb 28 2003

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