A196226 -id:A196226 - cp's OEIS frontend

A197638 GA1 numbers: composite m with G(m) >= G(m/p) for all prime factors p of m, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

4, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, Dec 02 2011

nonn

The members with exactly two prime divisors counted with multiplicity are 4 and 2*p, for primes p > 5. (See Section 5 of "Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis".)
The smallest member with more than two prime factors is 183783600. Such GA1 numbers are called "proper" - see A201557 and "Table of proper GA1 numbers up to 10^60".
The smallest odd member is 1058462574572984015114271643676625.
See "On SA, CA, and GA numbers".

			4 is a member because G(4) > 0 > G(2) = G(4/2).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010 [math.NT], 2011-2012.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.

Cf. A000203, A196226, A197639, A201557, A201558.

Maple
```
See "Computation of GA1 numbers".
```

g[k_] := g[k] = DivisorSigma[1, k]/(k*Log[Log[k]]); okQ[n_] := Module[{p = Transpose[FactorInteger[n]][[1]]}, i = 1; While[i <= Length[p] && g[n] >= g[n/p[[i]]], i++]; i > Length[p]]; Select[Range[2, 1000], ! PrimeQ[#] && okQ[#] &] (* T. D. Noe, Dec 03 2011 *)

g(k) = sigma(k)/(k*log(log(k)));
isga1(k)=if (isprime(k), return (0)); gk = g(k);f = factor(k); for(i=1,length(f~), if (gk < g(k/f[i,1]), return(0)););1; \\ Michel Marcus, Sep 09 2012

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 4, 2, 8, 2, 4, 8, 5, 2, 7, 2, 9, 8, 4, 2, 15, 3, 4, 5, 10, 2, 15, 2, 7, 8, 4, 8, 17, 2, 4, 8, 15, 2, 15, 2, 10, 14, 4, 2, 22, 3, 7, 8, 9, 2, 10, 8, 15, 8, 4, 2, 38, 2, 4, 14, 8, 8, 15, 2, 9, 7, 16, 2, 27, 2, 4, 13, 9, 8, 15, 2, 22, 6, 4, 2, 39, 8, 4, 7, 16, 2, 27, 8, 10

Offset: 3

Ctibor O. Zizka, Sep 01 2023

nonn

The equation y^2 = n^3 + A*n^2 + B*n + C, where A = 1, B = 2*r, C = r^2 is a minimal model of an elliptic curve with integral coefficients, for details see the Links section.
For a prime number p >= 5, the equation y^2 = p^3 + (p + r)^2 has 2 solutions, r_1 = p*(p - 3)/2 and r_2 = (p + 1)*(p^2 - p - 1)/2.
Factoring the equation y^2 = n^3 + n^2 + 2*r*n + r^2 yields (y+n+r)*(y-n-r) = n^3, which implies y+n+r = d and y-n-r = n^3/d for some divisor d of n^3.  Thus a(n) is the number of divisors d of n^3 such that (d-n^3/d)/2 - n is a nonnegative integer.  This resolves some of Thomas Scheuerle's conjectures. - Robin Visser, Sep 30 2023

			n = 6: y^2 = 6^3 + (6 + r)^2 is valid for r = 9, 19, 47, thus a(6) = 3. The 3 solutions [y, n, n+r] are [21, 6, 15], [29, 6, 25], [55, 6, 53].

Robin Visser, Table of n, a(n) for n = 3..10000
Joseph H. Silverman, The Arithmetic of Elliptic Curves

Cf. A081119, A134108, A190300, A196226, A245685, A364248.

a(n) = length(select((x) -> x[1] >= 0 && x[2] >= n, thue(thueinit(x^2-1,1),n^3),1)) \\ Thomas Scheuerle, Sep 03 2023

def a(n):
    num_sols = 0
    for d in Integer(n^3).divisors():
        if ((d-n^3/d)%2 == 0) and ((d-n^3/d)/2 >= n): num_sols += 1
    return num_sols  # Robin Visser, Sep 30 2023

a(p) = 2 for p prime >= 5, see Comments.
From Thomas Scheuerle, Sep 04 2023: (Start)
Conjecture: a(A190300(n)) = 3.
Conjecture: a(A196226(n)) = 4.
Conjecture: a(p^3) = 5 if p is an odd prime.
Conjecture: a(2*p^2) = 7 if p is an odd prime. But there exist other cases too, for example a(3*23) = 7.
Conjecture: a(prime(n)^prime(n)) = A245685(n - 1) - 1. (End)

a(61)-a(92) from Thomas Scheuerle, Sep 01 2023

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A197638 GA1 numbers: composite m with G(m) >= G(m/p) for all prime factors p of m, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

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A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.

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cp's OEIS Frontend

A197638 GA1 numbers: composite m with G(m) >= G(m/p) for all prime factors p of m, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

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Maple

Mathematica

PARI

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2*r*n + r^2.

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PARI

Sage

Formula

Extensions

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.