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%I #8 Aug 09 2015 00:30:02
%S 1,2,11,13,17,19,23,26,29,31,37,38,41,43,46,47,53,58,59,61,62,67,71,
%T 73,74,79,82,83,86,89,94,97,101,103,106,107,109,113,118,122,127,131,
%U 134,137,139,142,143,146,149,151,157,158,163,166,167,173,178,179,181,187
%N Numbers simultaneously heptagon-free, pentagon-free, squarefree and triangle-free.
%C 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}.
%H J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000.</a>
%F 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.
%F A113544 INTERSECT A113619. - _R. J. Mathar_, Jul 24 2009
%p 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
%t 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.
%Y Cf. A000217, A000566, A005117, A113502, A013929, A046098, A059956, A065474, A071172, A087618, A088454, A112886, A113508.
%K easy,nonn
%O 1,2
%A _Jonathan Vos Post_, Jan 14 2006
%E More terms from _R. J. Mathar_, Apr 19 2008
%E Extended by _R. J. Mathar_, Jul 24 2009
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