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

1,2

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 (i.e. 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

Table of n, a(n) for n=1..60.

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

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

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 <<NumberTheory`) determines whether a number is squarefree.

Crossrefs

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

Sequence in context: A068972 A116437 A048867 * A154539 A137238 A048521

Adjacent sequences: A113623 A113624 A113625 * A113627 A113628 A113629

Keyword

easy,nonn

Author

Jonathan Vos Post, Jan 14 2006

Extensions

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

Extended by R. J. Mathar, Jul 24 2009

Status

approved