A259366 -id:A259366 - cp's OEIS frontend

A060683 Numbers for which the differences between consecutive divisors (ordered by size) are distinct.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 95, 97, 98

Offset: 1

Labos Elemer, Apr 19 2001

nonn

A060682(a(n)) = A000005(a(n)) - 1, n > 1. - Reinhard Zumkeller, Jun 25 2015

			For n=6, divisors={1,2,3,6}; differences={1,1,3}, which are not distinct, so 6 is not in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. Balog, P. Erdős, and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.

Cf. A000005, A060680, A060681, A060682, A060765.

Cf. A060682, A259366 (complement).

a060683 n = a060683_list !! (n-1)
a060683_list = 1 : filter (\x -> a060682 x == a000005' x - 1) [2..]
-- Reinhard Zumkeller, Jun 25 2015

test[n_ ] := Length[dd=Drop[d=Divisors[n], 1]-Drop[d, -1]]==Length[Union[dd]]; Select[Range[1, 100], test]

isok(k) = my(d=divisors(k)); #Set(vector(#d-1, k, d[k+1]-d[k])) == #d-1; \\ Michel Marcus, Nov 11 2023

Edited by Dean Hickerson, Jan 22 2002

A129512 Numbers with at least two pairs of distinct divisors having equal differences.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 138, 140, 144, 150, 153, 156, 160, 162, 165, 168, 174, 176, 180, 182, 186, 189, 190, 192, 195

Offset: 1

Reinhard Zumkeller, Apr 19 2007

nonn

			See example for a(12) = 45 in A129510.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A129510, A066446, A129511 (complement).

Subsequences: A008588, A008597, A129521, A259366.

import Data.List.Ordered (minus)
a129512 n = a129512_list !! (n-1)
a129512_list = minus [1..] a129511_list
-- Reinhard Zumkeller, Aug 10 2015

q[k_] := Count[Tally[Differences /@ Subsets[Divisors[k], {2}] // Flatten][[;; , 2]], ?(# > 1 &)] > 0; Select[Range[200], q] (* _Amiram Eldar, Jan 27 2025 *)

is(n)=my(d=divisors(n)); for(i=1,#d-2, for(j=i+1,#d-1, for(k=1,#d, if(i!=k && setsearch(d, d[j]-d[i]+d[k]), return(1))))); 0 \\ Charles R Greathouse IV, Aug 26 2015

A129510(a(n)) < A066446(a(n)).

cp's OEIS Frontend

A060683 Numbers for which the differences between consecutive divisors (ordered by size) are distinct.

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