login
A113543
Numbers both squarefree and triangle-free.
1
1, 2, 5, 7, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 94, 95, 97, 101, 103, 106, 107, 109, 113, 115, 118, 119, 122, 127, 131, 133, 134, 137, 139, 142, 143, 145
OFFSET
1,2
COMMENTS
The cardinality (count, enumeration) of these through n equals n - card{squarefree numbers <= n} - card{trianglefree numbers <= n} + card{numbers <= n which are both square and triangular} = n - card{numbers <= n in A005117} - card{numbers <=n in A112886} + card{numbers <= n in A001110}. "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer. In fact, this problem may be no easier than the general problem of integer factorization (obviously, if an integer can be factored completely, is squarefree iff it contains no duplicated factors). This problem is an important unsolved problem in number theory" [Weisstein]. Conjecture: there is no polynomial time algorithm for recognizing numbers which are both squarefree and triangle-free.
REFERENCES
Bellman, R. and Shapiro, H. N. "The Distribution of Squarefree Integers in Small Intervals." Duke Math. J. 21, 629-637, 1954.
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, 2003.
Hardy, G. H. and Wright, E. M. "The Number of Squarefree Numbers." Section 18.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 269-270, 1979.
LINKS
Eric Weisstein's World of Mathematics, Squarefree.
FORMULA
a(n) has no factor >1 of form a*(a+1)/2 nor b^2. A005117 INTERSECTION A112886.
MATHEMATICA
bad = Rest@Union[# (# + 1)/2 &@ Range[19], Range[14]^2]; Select[ Range[200], {} == Intersection[bad, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jan 13 2006
EXTENSIONS
Corrected and extended by Giovanni Resta, Jun 13 2016
STATUS
approved