cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

Original entry on oeis.org

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

Views

Author

Matthew Vandermast, Feb 28 2003

Keywords

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • 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 *)

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.

A080998 a(n) is n's rank among the positive integers in terms of centrality -the fraction of n represented by the average gcd of n and the positive integers, or A018804(n)/n^2 (cf. A080997).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 7, 11, 9, 18, 8, 20, 13, 12, 15, 25, 14, 33, 16, 21, 23, 40, 17, 32, 28, 27, 22, 53, 19, 56, 30, 35, 39, 36, 24, 69, 46, 43, 26, 79, 29, 89, 38, 37, 55, 95, 31, 67, 45, 57, 47, 108, 41, 60, 42, 65, 75
Offset: 1

Views

Author

Matthew Vandermast, Mar 02 2003

Keywords

Examples

			a(5)=6 because the centrality of 5 is 9/25 (.36), which places it sixth among positive integers in centrality; it ranks behind 1 (whose centrality is 1), 2 (3/4, .75), 3 (5/9, .555...), 4 (1/2, .5) and 6 (5/12, .41666...).

Crossrefs

A080997 gives fuller description of centrality, as well as finite portion of the arrangement of positive integers in order of their centrality. For more about where numbers occur in A080997, see also A081000, A081001, A081028, A081029.

A081001 n is in the sequence if and only if it does not rank among the top n positive integers in centrality (cf. A080997 for fuller explanation of this concept).

Original entry on oeis.org

5, 7, 9, 11, 13, 17, 19, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 116, 118, 119
Offset: 1

Views

Author

Matthew Vandermast, Mar 02 2003

Keywords

Comments

Complement of A081000.

Crossrefs

Cf. A080997, A080998, also A081028 for centrality ranks of primes (all of which are members of this sequence except 2 and 3).

Formula

Formula for the centrality of n: A018804(n)/n^2 (see also A080997).

A081029 Highly central numbers: numbers having a centrality higher than that of any larger number.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 18, 20, 24, 30, 36, 40, 42, 48, 60, 72, 84, 90, 120, 126, 144, 180, 210, 240
Offset: 1

Views

Author

Matthew Vandermast, Mar 02 2003

Keywords

Comments

A subset of A081000.

Crossrefs

Cf. A080997, A080998 for centrality rankings of the positive integers.

Formula

Formula for centrality of n: A018804(n)/n^2 (cf. A080997 for fuller description of this concept).
Showing 1-4 of 4 results.

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