A097703 -id:A097703 - 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

A097702 a(n) = (A063880(n) - 108)/216.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 11, 14, 15, 17, 18, 20, 21, 23, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 86, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 110

Offset: 1

Ralf Stephan, Aug 26 2004

nonn

Conjecture: k is a term iff 6*k+3 is squarefree. - Vladeta Jovovic, Aug 27 2004
It is only a conjecture that all terms are integers (confirmed up to 10^6 by Robert G. Wilson v).
From Amiram Eldar, Aug 31 2024: (Start)
The first conjecture is true. If m = 216*k + 108 = 108 * (2*k + 1) is a term of A063880, then 2*k+1 is a squarefree number coprime to 6. This is because sigma(n)/usigma(n) is multiplicative, equals 1 if and only if n is squarefree and larger than 1 otherwise, sigma(108)/usigma(108) = 2 and sigma(3^k)/usigma(3^k) increases with k. 6*k+3 = 3*(2*k+1) is squarefree because 2*k+1 is a squarefree coprime to 6.
Assuming that (A063880(n) - 108)/216 is an integer for all n, we have a(n) = (A276378(n) - 1)/2. (End)

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

Cf. A000203, A005117, A034448, A063880, A097703, A276378.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; (Select[ Range[ 24500], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216 (* Robert G. Wilson v, Aug 28 2004 *)

{u(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d))}
n=2; while(n<50000,n++; if(sigma(n)==2*u(n),print1((n-108)/216", ")))

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

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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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A097702 a(n) = (A063880(n) - 108)/216.

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