OFFSET
1,1
COMMENTS
Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - Reinhard Zumkeller, Mar 10 2003
The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - Labos Elemer, Nov 25 2002
1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - Amarnath Murthy, Apr 24 2003
The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - Amiram Eldar, Feb 14 2021
Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - Gus Wiseman, Sep 14 2024
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.
FORMULA
A261869(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2015
EXAMPLE
44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
From Gus Wiseman, Sep 14 2024: (Start)
Splitting nonsquarefree numbers into maximal anti-runs gives:
(4,8)
(9,12,16,18,20,24)
(25,27)
(28,32,36,40,44)
(45,48)
(49)
(50,52,54,56,60,63)
(64,68,72,75)
(76,80)
(81,84,88,90,92,96,98)
(99)
The maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
(End)
MATHEMATICA
Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000, n, 1], Table[False, {n}]]; f@2 (* Hans Rudolf Widmer, Aug 30 2022 *)
Max/@Split[Select[Range[100], !SquareFreeQ[#]&], #1+1!=#2&]//Most (* Gus Wiseman, Sep 14 2024 *)
PROG
(Haskell)
a068781 n = a068781_list !! (n-1)
a068781_list = filter ((== 0) . a261869) [1..]
-- Reinhard Zumkeller, Sep 04 2015
(PARI) isok(m) = !moebius(m) && !moebius(m+1); \\ Michel Marcus, Feb 14 2021
CROSSREFS
Subsequence of A261869.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Mar 04 2002
STATUS
approved