A068781 Lesser of two consecutive numbers each divisible by a square.

8, 24, 27, 44, 48, 49, 63, 75, 80, 98, 99, 116, 120, 124, 125, 135, 147, 152, 168, 171, 175, 188, 207, 224, 242, 243, 244, 260, 275, 279, 288, 296, 315, 324, 332, 342, 343, 350, 351, 360, 363, 368, 375, 387, 404, 423, 424, 440, 459, 475, 476, 495, 507, 512

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

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.

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 *)

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

Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.

Cf. A049535, A077647, A078143, A045882.

Subsequence of A261869.

Sequence in context: A176297 A175496 A048109 * A365864 A212861 A333961

Adjacent sequences: A068778 A068779 A068780 * A068782 A068783 A068784

Keyword

nonn

Author

Robert G. Wilson v, Mar 04 2002

Status

approved