A097704 -id:A097704 - cp's OEIS frontend

A067778 Numbers k such that gcd(k, numerator(B(2k))) is not squarefree where B(2k) are the Bernoulli numbers.

25, 49, 75, 98, 121, 125, 169, 175, 196, 225, 242, 245, 275, 289, 325, 338, 343, 361, 363, 375, 392, 425, 475, 484, 490, 507, 525, 529, 539, 575, 578, 625, 637, 675, 676, 686, 722, 725, 726, 775, 784, 825, 833, 841, 845, 847, 867, 875, 925, 931, 961, 968

Offset: 1

Benoit Cloitre, Feb 06 2002

nonn

All members so far are themselves not squarefree. - Robert G. Wilson v, Aug 28 2004

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

Cf. A013929, A027641, A027642, A097704.

Select[ Range[1000], !SquareFreeQ[ GCD[ #, Numerator[ BernoulliB[2# ]] ]] &] (* Robert G. Wilson v, Aug 28 2004 *)

More terms from Robert G. Wilson v, Aug 28 2004

A097703 Numbers j such that m = 216j + 108 satisfies sigma(m) != 2usigma(m).

Original entry on oeis.org

1, 4, 7, 10, 12, 13, 16, 19, 22, 24, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 60, 61, 62, 64, 67, 70, 73, 76, 79, 82, 84, 85, 87, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 122, 124, 127, 130, 133, 136, 137, 139, 142, 144, 145, 148, 151, 154, 157

Offset: 1

Ralf Stephan, Aug 26 2004

nonn

Conjecture: all numbers of form 3k + 1 are here. Other terms are listed in A097704.
From Amiram Eldar, Aug 31 2024: (Start)
The conjecture is true. If j = 3*k+1, then m = 324*(2*k+1). Let e = A007949(2*k+1) >= 0, so 2*k+1 = 3^e * i and i coprime to 6. Then sigma(m)/(2 * usigma(m)) = (7/20) * (3^(e+5)-1)/(3^(e+4)+1) * sigma(i)/usigma(i) >= 847/820 > 1, because sigma(i)/usigma(i) >= 1 for all i.
If m = 216*j + 108 = 108*(2*j+1) then sigma(m) = 2*usigma(m) if and only if 2*j+1 is a squarefree number coprime to 3 (see A097702), i.e., 2*j+1 is a term of A276378. Therefore this sequence consists of numbers j such that 2*j+1 is either a multiple of 3 or nonsquarefree (or both). (End)

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

Complement of A097702.

Cf. A000203, A007949, A013929, A034448, A063880, A097704, A276378.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Complement[ Range[157], (Select[ Range[37000], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216] (* Robert G. Wilson v, Aug 28 2004 *)

is(k) = {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

A067778 Numbers k such that gcd(k, numerator(B(2k))) is not squarefree where B(2k) are the Bernoulli numbers.

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A097703 Numbers j such that m = 216j + 108 satisfies sigma(m) != 2usigma(m).

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

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

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cp's OEIS Frontend

A067778 Numbers k such that gcd(k, numerator(B(2*k))) is not squarefree where B(2*k) are the Bernoulli numbers.

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A097703 Numbers j such that m = 216*j + 108 satisfies sigma(m) != 2*usigma(m).

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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

A067778 Numbers k such that gcd(k, numerator(B(2k))) is not squarefree where B(2k) are the Bernoulli numbers.

A097703 Numbers j such that m = 216j + 108 satisfies sigma(m) != 2usigma(m).