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A046790
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Positive numbers divisible by 8 or by the square of an odd prime.
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14
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8, 9, 16, 18, 24, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 88, 90, 96, 98, 99, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 135, 136, 144, 147, 150, 152, 153, 160, 162, 168, 169, 171, 175, 176, 180, 184, 189, 192, 196, 198, 200, 207, 208
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OFFSET
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1,1
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COMMENTS
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This sequence has many equivalent definitions:
(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.)
(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
(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
For a proof of the equivalence of definitions (D1)-(D5) see the Don Reble link.
(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
(D7) Numbers that can be the semiperimeter of a isosceles triangle with integer sides and area. - Peter Kagey, May 17 2019
Closed under multiplication, which may be used to construct the sequence. - David A. Corneth, Jun 07 2016
m is in this sequence if and only if m does not divide 2*radical(m). - Peter Luschny, Mar 05 2019
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LINKS
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Emmanuel Amiot, Autosimilar Melodies, J. Math. Music 2 (2008), no. 3, 157-180. DOI: 10.1080/17459730802598146.
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FORMULA
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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
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MATHEMATICA
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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 *)
Select[Range[210], Mod[#, 8] == 0 || AnyTrue[ Divisors[#], DivisorSigma[0, #] == 3 && Mod[#, 4] != 0 &] &] (* Carlos Eduardo Olivieri, Jun 07 2016 *)
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 *)
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PROG
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(PARI) is(n)={n%8==0||!issquarefree(n>>!bittest(n, 0))} \\ M. F. Hasler, Jun 07 2016
(Sage) print([n for n in (1..208) if not ZZ(n).divides(2*radical(n))]) # Peter Luschny, Mar 05 2019
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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