A077223 - OEIS

%I #22 Sep 04 2023 22:36:57

%S 1,2,3,4,7,6,5,8,9,10,11,12,17,14,15,16,13,18,19,20,21,22,25,26,27,24,

%T 23,28,29,30,31,34,33,32,35,36,37,40,39,38,41,42,43,44,45,46,47,48,49,

%U 52,51,50,53,54,55,56,57,58,61,62,65,64,59,60,63,66,67,70,69,68,71,72

%N a(1) = 1 and then alternately even and odd numbers not occurring earlier such that the sum of any two successive terms is a squarefree number.

%C No string of 9 successive terms can be consecutive integers simply because one of the sums of pair of terms would be a multiple of 9=3^2.

%C Conjecture: There are infinitely many strings of 8 terms of the form k, k+1, k+2, ..., k+7.

%C Possible subsidiary sequences: Start of the strings of 8 consecutive natural numbers. Start of the strings of r consecutive natural numbers for a particular r, 3 < r < 8.

%C A self-inverse permutation of the natural numbers.

%H Reinhard Zumkeller, <a href="/A077223/b077223.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%o (Haskell)

%o import Data.List  (find, delete)

%o import Data.Maybe (fromJust)

%o a077223 n = a077223_list !! (n-1)

%o a077223_list = 1 : g 1 [2..] where

%o    g i xs = x : g x (delete x xs) where

%o             x = (fromJust $ find isOddSquarefree $ map (+ i)

%o    isOddSquarefree m = odd m && a008966 m == 1

%o for_bFile = take 10000 a077223_list

%Y Cf. A008966.

%K nice,nonn

%O 1,2

%A _Amarnath Murthy_, Nov 03 2002

%E a(16)-a(71) from _Donovan Johnson_, Nov 17 2008