A201557 -id:A201557 - cp's OEIS frontend

A216436 For g=A201557(n), define a(n) as the prime p|g such that G(g/p) is maximum, where G(k) = sigma(k)/(k*log(log(k))).

17, 17, 19, 19, 19, 19, 19, 23, 23, 31, 31, 31, 7, 7, 7, 7, 7, 37, 37, 41, 43, 43, 43, 43, 43, 43, 2, 43, 2, 43, 2, 47, 47, 47, 47, 47, 47, 2, 47, 2, 47, 2, 47, 2, 53, 61, 61, 61, 61, 61, 61, 5, 5, 5, 67, 71, 11, 11, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73

Offset: 1

Michel Marcus, Sep 10 2012

nonn

The ratio G(A201557(n)/p)/G(A201557(n)) is defined by Nicolas as the Gronwall quotient (A201557 = proper GA1 numbers).

Michel Marcus, Table of n, a(n) for n = 1..1000
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.
J.-L. Nicolas, Computation of GA1 numbers, 2011.

Cf. A000203, A201557, A197638.

# See link "Computation of GA1 numbers".

findq(i) = {f = factor(i); maxqu = 0.0; qmax = 0; for(iq=1, length(f~), qq = f[iq,1]; qu = g(i/qq)/g(i); if (qu > maxqu, maxqu = qu; qmax = qq;) ); return (qmax);} \\ for i in A201557

G(A201557(n)/a(n)) >= G(A201557(n)/q) if prime q|A201557(n).

Definition simplified and formula supplied by Jonathan Sondow, Sep 11 2012

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.

Original entry on oeis.org

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

A197639 GA2 numbers: n with G(n) >= G(an) for all integers a > 0, where G(k) = sigma(k)/(klog(log(k))) and sigma(k) = sum of divisors of k.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 18, 24, 36, 48, 60, 72, 120, 180, 240, 360, 2520, 5040

Offset: 1

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

nonn

Subsequence of A067698.

A member > 5040 exists iff the Riemann Hypothesis is false, in which case the sequence is infinite. In any case, 3 and 5 are the only odd members. (See Sections 1 and 4 of "On SA, CA, and GA numbers".)

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, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Cf. A000203, A067698, A197638, A201557.

nmax = 10^6; amax = 10;
G[k_] := DivisorSigma[1, k]/(k Log[Log[k]]);
okQ[n_] := DivisorSigma[1, n] > n Exp[EulerGamma] Log[Log[n]] && AllTrue[ Range[amax], Function[a, G[n] >= G[a*n]]];
Reap[For[n = 1, n <= nmax, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 30 2019 *)

A201558 Number of GA1 numbers A197638 with n >= 3 prime factors counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 1, 1, 2, 4, 1, 2, 3, 7, 7, 7, 1, 4, 7

Offset: 3

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 03 2011

nonn

The number of GA1 numbers with one (resp., two) prime factors is zero (resp., infinity).
GA1 numbers with at least three prime factors are called "proper" - see A201557.
For a(n), see Section 6.2 of "On SA, CA, and GA numbers", and below "kmax" in "Table of proper GA1 numbers up to 10^60".

			183783600 = 2^4 * 3^3 * 5^2 * 7 * 11 * 13 * 17 is the first of the a(13) = 2 GA1 numbers with 4 + 3 + 2 + 1 + 1 + 1 + 1 = 13 prime factors.

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.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.

Cf. A067698, A197638, A197639, A201557.

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

cp's OEIS Frontend

A216436 For g=A201557(n), define a(n) as the prime p|g such that G(g/p) is maximum, where G(k) = sigma(k)/(k*log(log(k))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A197639 GA2 numbers: n with G(n) >= G(a*n) for all integers a > 0, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A201558 Number of GA1 numbers A197638 with n >= 3 prime factors counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A197639 GA2 numbers: n with G(n) >= G(an) for all integers a > 0, where G(k) = sigma(k)/(klog(log(k))) and sigma(k) = sum of divisors of k.