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%I #140 Jul 27 2023 12:10:15
%S 8,9,16,18,24,25,27,32,36,40,45,48,49,50,54,56,63,64,72,75,80,81,88,
%T 90,96,98,99,100,104,108,112,117,120,121,125,126,128,135,136,144,147,
%U 150,152,153,160,162,168,169,171,175,176,180,184,189,192,196,198,200,207,208
%N Positive numbers divisible by 8 or by the square of an odd prime.
%C This sequence has many equivalent definitions:
%C (D1) Positive numbers divisible by 8 or by the square of an odd prime. (We take this as the main definition, since it is the simplest.)
%C (D2) Moduli m for which there exist affine maps f:x->a*x + b modulo m, with a > 1, such that f has order m in the affine group. (For example, 8 is a term because f:x->(5x+1) mod 8 is a map with order 8 in the group of affine maps mod 8: the smallest power of f equal to identity is f^8. The maps x->x+1 always have this property, so are excluded from consideration.) - _Emmanuel Amiot_, Jul 28 2007
%C (D3) Numbers k such that A005361(k) < A003557(k). - _Anthony Browne_, Jun 03 2016
%C (D4) Numbers i such that there is a smaller positive number j such that (i+j)/2 and sqrt(i*j) are integers. (See A046791 for the smallest choice for j.) - _David W. Wilson_, Dec 11 1999
%C (D5) Numbers k such that A008475(k) is different from A001414(k). - _Benoit Cloitre_, Jan 11 2003
%C For a proof of the equivalence of definitions (D1)-(D5) see the Don Reble link.
%C (D6) Numbers m >= 8 having a divisor k^2 >= 4 such that m and m/k^2 are of the same parity. (See A046791 for the largest such k.) - _Vladimir Shevelev_, Jun 06 2006
%C (D7) Numbers that can be the semiperimeter of a isosceles triangle with integer sides and area. - _Peter Kagey_, May 17 2019
%C Closed under multiplication, which may be used to construct the sequence. - _David A. Corneth_, Jun 07 2016
%C Complement of A078779. - _Omar E. Pol_, Jun 11 2016
%C m is in this sequence if and only if m does not divide 2*radical(m). - _Peter Luschny_, Mar 05 2019
%H M. F. Hasler, <a href="/A046790/b046790.txt">Table of n, a(n) for n = 1..10000</a> (first 290 terms from N. J. A. Sloane).
%H Emmanuel Amiot, <a href="http://dx.doi.org/10.1080/17459730802598146">Autosimilar Melodies</a>, J. Math. Music 2 (2008), no. 3, 157-180. DOI: 10.1080/17459730802598146.
%H Emmanuel Amiot, <a href="http://canonsrythmiques.free.fr/pdf/maMuXMelodieAmiot.pdf">Mélodies autosimilaires (Self-Replicating Melodies)</a> (in French).
%H Don Reble, <a href="/A046790/a046790_1.txt">Proof of equivalence of definitions (D1)-(D5)</a>, Jun 06 2016
%H Mathematics Stack Exchange user "George", <a href="https://math.stackexchange.com/a/3230072/121988">Semiperimeter of isosceles Heronian triangles</a>.
%F Let A(x) be the number of a(n) <= x. Then A(x) ~ (1 - 7/Pi^2)*x = 0.2907517...*x as x goes to infinity. - _Vladimir Shevelev_, Jun 07 2016
%t ordreMax[a_, n_]:= Module[{mo, r, s, s0, gcd}, mo=MultiplicativeOrder[a,n]; s= s0=Mod[Sum[a^k,{k,0,mo-1}], n]; Max[Table[gcd=GCD[a-1,b];r=1; While[Mod[s *gcd, n]!=0, s=Mod[s0+a^mos,n];r++ ]; r*mo, {b,0,n-1} ]] ] ordreMax[n_] := Module[{candidats, m,t}, candidats = Select[Range[2,n-1], (GCD[n,# ]==1 && GCD[n, #-1]>1)&]; m=Max[t=Table[ordreMax[a,n], {a, candidats}] ]; {m,Part[candidats,Flatten@Position[t, m] ]}] Module[{resu}, Do[resu=ordreMax[n]; If[First[resu] >=n, Print[n ]], {n,4,200}]] (* This is for definition (D2). _Emmanuel Amiot_, Jul 28 2007 *)
%t Select[Range[210], Mod[#, 8] == 0 || AnyTrue[ Divisors[#], DivisorSigma[0, #] == 3 && Mod[#, 4] != 0 &] &] (* _Carlos Eduardo Olivieri_, Jun 07 2016 *)
%t Module[{upto=250,prs},prs=Prime[Range[2,PrimePi[Sqrt[upto]]]]^2;Join[ Range[ 8,upto,8],Select[Range[upto],AnyTrue[#/prs,IntegerQ]&]]] // Union (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 18 2018 *)
%o (PARI) is(n)={n%8==0||!issquarefree(n>>!bittest(n,0))} \\ _M. F. Hasler_, Jun 07 2016
%o (Sage) print([n for n in (1..208) if not ZZ(n).divides(2*radical(n))]) # _Peter Luschny_, Mar 05 2019
%Y Cf. A001414, A003557, A005361, A008475, A046791.
%K nonn,nice
%O 1,1
%A _David W. Wilson_, Dec 11 1999
%E Entry revised by _N. J. A. Sloane_, with help from _Don Reble_ and several OEIS editors, Jun 07 2016
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