A046790 Positive numbers divisible by 8 or by the square of an odd prime.

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

Offset: 1

Author

David W. Wilson, Dec 11 1999

Keywords

• nonn
• nice

Comments

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

(D3) Numbers k such that A005361(k) < A003557(k). - Anthony Browne, Jun 03 2016

(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

(D5) Numbers k such that A008475(k) is different from A001414(k). - Benoit Cloitre, Jan 11 2003

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

Complement of A078779. - Omar E. Pol, Jun 11 2016

m is in this sequence if and only if m does not divide 2*radical(m). - Peter Luschny, Mar 05 2019

Verified up to a(290) = 1000, {a(n)} is identical to the sequence of group orders for which there exists at least one group G such that |Char(G)| is a nontrivial divisor of |Normal(G)|, where |Char(G)| is the number of characteristic subgroups of G and |Normal(G)| the number of normal subgroups of G. - Miles Englezou, Jul 20 2024

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 290 terms from N. J. A. Sloane).
• Emmanuel Amiot, Autosimilar Melodies, J. Math. Music 2 (2008), no. 3, 157-180. DOI: 10.1080/17459730802598146.
• Emmanuel Amiot, Mélodies autosimilaires (Self-Replicating Melodies) (in French).
• Don Reble, Proof of equivalence of definitions (D1)-(D5), Jun 06 2016
• Mathematics Stack Exchange user "George", Semiperimeter of isosceles Heronian triangles.

Crossrefs

Cf. A001414, A003557, A005361, A008475, A046791.

Programs

Mathematica

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 *)

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

Formula

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

Extensions

Entry revised by N. J. A. Sloane, with help from Don Reble and several OEIS editors, Jun 07 2016