A067778 -id:A067778 - cp's OEIS frontend

A097704 Terms of A097703 that are not of the form 3*k + 1.

12, 24, 60, 62, 84, 87, 122, 137, 144, 162, 171, 180, 212, 237, 264, 269, 287, 302, 312, 318, 362, 387, 416, 420, 422, 423, 437, 462, 465, 480, 512, 537, 563, 587, 591, 612, 662, 665, 684, 687, 710, 722, 737, 759, 762, 786, 812, 837, 840, 857, 887, 902

Offset: 1

Ralf Stephan, Aug 26 2004

nonn

Conjecture: "most" of the terms also belong to [(A067778-1)/2]. Exceptions are {302, 2117, ...} (A098241). In other words, most terms satisfy: GCD(2*k+1, numerator(B(4*k+2))) is not squarefree, with B(n) the Bernoulli numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A007494 and A097703.

Cf. A013929, A027641, A027642, A067778, A098241.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Complement[ Range[1017], Table[3k - 2, {k, 340}], (Select[ Range[220000], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216] (* Robert G. Wilson v, Aug 28 2004 *)

is(k) = if(k % 3 == 1, 0, my(f = factor(216*k + 108)); sigma(f) != 2 * prod(i = 1, #f~, 1 + f[i,1]^f[i,2])); \\ Amiram Eldar, Aug 31 2024

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

Original entry on oeis.org

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

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

nonn

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.

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

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

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 *)

More terms from Amiram Eldar, Aug 31 2024

A098242 Numbers k such that gcd(2k+1, numerator(Bernoulli(4k+2))) is not squarefree.

Original entry on oeis.org

12, 24, 37, 60, 62, 84, 87, 112, 122, 137, 144, 162, 171, 180, 181, 187, 212, 237, 253, 262, 264, 269, 287, 312, 318, 337, 362, 387, 412, 416, 420, 422, 423, 433, 437, 462, 465, 480, 487, 512, 537, 541, 544, 562, 563, 587, 591, 612, 637, 662, 665, 684, 687

Offset: 1

Ralf Stephan, Sep 01 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Complement of A098240.

Cf. A013929, A027641, A027642, A067778, A097704.

Select[Range[700], !SquareFreeQ[GCD[2*# + 1, BernoulliB[4*# + 2]]] &] (* Amiram Eldar, Aug 31 2024 *)

for(n=1,3000,if(!issquarefree(gcd(2*n+1,numerator(bernfrac(4*n+2)))),print1(n",")))

cp's OEIS Frontend

A097704 Terms of A097703 that are not of the form 3*k + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A098242 Numbers k such that gcd(2k+1, numerator(Bernoulli(4k+2))) is not squarefree.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI