This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A113543 #18 Feb 16 2025 08:32:59
%S A113543 1,2,5,7,11,13,14,17,19,22,23,26,29,31,34,35,37,38,41,43,46,47,53,58,
%T A113543 59,61,62,65,67,71,73,74,77,79,82,83,85,86,89,94,95,97,101,103,106,
%U A113543 107,109,113,115,118,119,122,127,131,133,134,137,139,142,143,145
%N A113543 Numbers both squarefree and triangle-free.
%C A113543 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.
%D A113543 Bellman, R. and Shapiro, H. N. "The Distribution of Squarefree Integers in Small Intervals." Duke Math. J. 21, 629-637, 1954.
%D A113543 Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, 2003.
%D A113543 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.
%H A113543 G. C. Greubel, <a href="/A113543/b113543.txt">Table of n, a(n) for n = 1..1000</a>
%H A113543 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree.</a>
%F A113543 a(n) has no factor >1 of form a*(a+1)/2 nor b^2. A005117 INTERSECTION A112886.
%t A113543 bad = Rest@Union[# (# + 1)/2 &@ Range[19], Range[14]^2]; Select[ Range[200], {} == Intersection[bad, Divisors[#]] &] (* _Giovanni Resta_, Jun 13 2016 *)
%Y A113543 Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886.
%K A113543 easy,nonn
%O A113543 1,2
%A A113543 _Jonathan Vos Post_, Jan 13 2006
%E A113543 Corrected and extended by _Giovanni Resta_, Jun 13 2016