

A113543


Numbers both squarefree and trianglefree.


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
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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 trianglefree.


REFERENCES

Bellman, R. and Shapiro, H. N. "The Distribution of Squarefree Integers in Small Intervals." Duke Math. J. 21, 629637, 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. 269270, 1979.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000
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 *)


CROSSREFS

Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886.
Sequence in context: A075610 A057922 A243047 * A189468 A004134 A191406
Adjacent sequences: A113540 A113541 A113542 * A113544 A113545 A113546


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Jan 13 2006


EXTENSIONS

Corrected and extended by Giovanni Resta, Jun 13 2016


STATUS

approved



