A113626 Numbers simultaneously heptagon-free, pentagon-free, squarefree and triangle-free.

1, 2, 11, 13, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 143, 146, 149, 151, 157, 158, 163, 166, 167, 173, 178, 179, 181, 187

Offset: 1

Author

Jonathan Vos Post, Jan 14 2006

Keywords

• easy
• nonn

Comments

This sequence is the 5th step in a polygonal-factor sieve, where all integers with k-gonal factors have been eliminated from an initial set of the natural numbers, for k = 3, 4, 5, .... There is no need to specifically sieve out hexagonal numbers, as every hexagonal number is a triangular number and thus is already sieved. Every integer n is sieved out no later than step n-3, as n-gonal number(2) = n (e.g. 7 is eliminated when we sieve out all numbers with heptagonal factors, as 7 = Hep(2); 11 is eliminated when we sieve out all 11-gonal number multiples). After an infinite number of steps, the sequence collapses to {1,2}. If, instead, at each step we eliminate all multiples of n-gonal numbers except {1, n} then the sequence converges on {1,4} UNION {primes}.

Links

• J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

Crossrefs

Cf. A000217, A000566, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.

Programs

Maple

isA000217 := proc(n) local discr ; discr := 1+8*n ; if issqr(discr) then if ( sqrt(discr)-1 ) mod 2 = 0 then true; else false ; fi ; else false ; fi ; end: isA000326 := proc(n) local discr ; discr := 1+24*n ; if issqr(discr) then if ( sqrt(discr)+1 ) mod 6 = 0 then true; else false ; fi ; else false ; fi ; end: isA000566 := proc(n) local discr ; discr := 9+40*n ; if issqr(discr) then if ( sqrt(discr)+3 ) mod 10 = 0 then true; else false ; fi ; else false ; fi ; end: isA000290 := proc(n) issqr(n) ; end: isA113626 := proc(n) local d ; for d in numtheory[divisors](n) do if d > 1 then if isA000217(d) or isA000290(d) or isA000326(d) or isA000566(d) then RETURN(false) ; fi ; fi ; od: RETURN(true) ; end: for n from 1 to 500 do if isA113626(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Apr 19 2008

Mathematica

The Mathematica function SquareFreeQ[n] in the Mathematica add-on package NumberTheory`NumberTheoryFunctions` (which can be loaded with the command <
				

Formula

a(n) has no factor >1 of form b*(b+1)/2, c^2, d*(3*d-1)/2, nor e*(5*e-3)/2.

A113544 INTERSECT A113619. - R. J. Mathar, Jul 24 2009

Extensions

More terms from R. J. Mathar, Apr 19 2008

Extended by R. J. Mathar, Jul 24 2009