OFFSET
1,1
COMMENTS
The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of triples of terms = {900a+548, 900a+549, 900a+550}=4(225f+137), 9(100f+61), 25(36f+22)}; starting terms in this sequence = {549, 1458, 2358, ...}; difference = A002110(3)^2. - Labos Elemer, Nov 25 2002
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 2, 16, 180, 1868, 18649, 186335, 1863390, 18634236, 186340191, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01863... . - Amiram Eldar, Jan 18 2023
The asymptotic density of this sequence is 1 - 3/zeta(2) + 3 * Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = 1 - 3 * A059956 + 3 * A065474 - A206256 = 0.018634010349844827414... . - Amiram Eldar, Sep 12 2024
REFERENCES
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris, 2008.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
FORMULA
a(n) = A235578(n) - 1. - Amiram Eldar, Feb 09 2021
MATHEMATICA
f[n_] := Union[ Transpose[ FactorInteger[n]] [[2]]] [[ -1]]; a = 0; b = 1; Do[c = f[n]; If[a> 1 && b > 1 && c > 1, Print[n - 2]]; a = b; b = c, {n, 3, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[3000], 3, 1], _?(Union[#] == {False}&), {1}, Heads->False]] (* Harvey P. Dale, May 24 2014 *)
f@n_ := Flatten@ Position[Partition[SquareFreeQ /@ Range@2000, n, 1], Table[False, {n}]]; f@3 (* Hans Rudolf Widmer, Aug 30 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002
EXTENSIONS
More terms from Jason Earls and Robert G. Wilson v, May 10 2002
Offset corrected by Amiram Eldar, Feb 09 2021
STATUS
approved