A098241 Numbers k such that 216*k+108 is a term of A097703 and A007494 and A098240.

302, 2117, 2909, 3327, 3932, 5142, 5747, 6957, 8772, 9377, 11192, 12402, 13007, 14217, 14547, 16032, 17847, 18452, 20267, 20366, 21477, 22082, 23292, 23897, 25107, 25403, 26922, 27527, 29342, 30552, 31157, 32367, 32972, 34182, 35997, 36602, 37823, 38417, 39627

Offset: 1

Author

Ralf Stephan and Robert G. Wilson v, Sep 15 2004

Keywords

• nonn

Comments

Numbers k such that m = 216*k+108 satisfies sigma(m) <> 2*usigma(m) (A097703), m is not of the form 3x+1 (A007494) and GCD(2*m+1, numerator(Bernoulli(4*m+2))) is squarefree (A098240).

Also, terms m of A097704 such that GCD(2*m+1, Bernoulli(4*m+2)) is squarefree. Most terms of A097704 are in A098240. These are the exceptions.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000203, A034448, A007494, A016777, A027641, A027642, A063880, A067778, A097703, A097704, A098240.

Programs

Mathematica

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; lmt = 1296000; t = (Select[ Range[ lmt], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216; u = (Select[ Range[ Floor[(lmt - 108)/432]], !SquareFreeQ[ GCD[ #, Numerator[ BernoulliB[ 2# ]] ]] &] -1)/2; v = Table[ 3k - 2, {k, Floor[(lmt - 108)/216]}]; Complement[ Range[ Floor[ (lmt - 108)/216]], t, u, v]
q[n_] := Mod[n, 3] != 1 && (Divisible[2*n + 1, 3] || (! Divisible[2*n + 1, 3] && ! SquareFreeQ[2*n + 1])) && SquareFreeQ[GCD[2*n + 1, BernoulliB[4*n + 2]]]; Select[Range[10^4], q] (* Amiram Eldar, Aug 31 2024 *)

Extensions

More terms from Amiram Eldar, Aug 31 2024