A113544 Numbers simultaneously pentagon-free, squarefree and triangle-free.

1, 2, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 77, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 119, 122, 127, 131, 133, 134, 137, 139, 142, 143, 146, 149, 151, 157, 158, 161, 163

Offset: 1

Author

Jonathan Vos Post, Jan 13 2006

Keywords

• easy
• nonn

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

• G. C. Greubel and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Greubel)
• Eric Weisstein's World of Mathematics, Squarefree.

Crossrefs

Cf. A000217, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.

Programs

Mathematica

bad = Rest@ Union[# (# + 1)/2 &@ Range[19], Range[14]^2, # (3 # - 1)/2 &@ Range[11]]; Select[Range[200], {} == Intersection[bad, Divisors[#]] &] (* Giovanni Resta, Jun 13 2016 *)

PARI

list(lim)=my(v=List()); forsquarefree(n=1,lim\1, fordiv(n,d, if((ispolygonal(d,3) || ispolygonal(d,5)) && d>1, next(2))); listput(v,n[1])); Vec(v); \\ Charles R Greathouse IV, Dec 24 2018

Formula

a(n) has no factor >1 of form a*(a+1)/2 nor b^2 nor c*(3*c-1)/2. A005117 INTERSECTION A112886 INTERSECTION A113508.

Extensions

Corrected and extended by Giovanni Resta, Jun 13 2016