A197639 -id:A197639 - 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

A201557 Proper GA1 numbers: terms of A197638 with at least three prime divisors counted with multiplicity.

Original entry on oeis.org

183783600, 367567200, 1396755360, 1745944200, 2327925600, 3491888400, 6983776800, 80313433200, 160626866400, 252706217563800, 288807105787200, 336941623418400, 404329948102080, 505412435127600, 673883246836800, 1010824870255200, 2021649740510400, 112201560598327200

Offset: 1

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

nonn

Infinitely many terms are superabundant (SA) A004394; the smallest is 183783600.
Infinitely many terms are colossally abundant (CA) A004490; the smallest is 367567200.
Infinitely many terms are odd (and hence neither SA nor CA); the smallest is 1058462574572984015114271643676625.
See Section 5 of "On SA, CA, and GA numbers".
For additional terms, in factored form, see "Table of proper GA1 numbers up to 10^60", where SA and CA numbers are starred * and **.

			183783600 = 2^4 * 3^3 * 5^2 * 7 * 11 * 13 * 17 is the smallest proper GA1 number.

Amiram Eldar, Table of n, a(n) for n = 1..18408 (from the J.-L. Nicolas's table)
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. A000203, A001222, A067698, A197638, A197639, A201558, A216436.

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

A197638(n) if A001222(A197638(n)) > 2

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

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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A201557 Proper GA1 numbers: terms of A197638 with at least three prime divisors counted with multiplicity.

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

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