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A322154 Numbers n satisfying gcd(n^2, sigma(n^2)) > sigma(n), where sigma is the sum-of-divisors function. 0
693, 1386, 1463, 1881, 2379, 2926, 4389, 4758, 8778, 9516, 11895, 13167, 16653, 18018, 19032, 23790, 24180, 25641, 26169, 26334, 33306, 37271, 40443, 43890, 45201, 52668, 54717, 57057, 61380, 65835, 73150, 78507, 105336, 109725, 111813, 114114, 131670, 157014, 166530, 169959 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let N = q^k*n^2 be an odd perfect number with special prime q.  If k = 1, it follows that sigma(q^k) < n.  Since 2n^2/sigma(q^k) = gcd(n^2, sigma(n^2)), if k = 1 then we have gcd(n^2, sigma(n^2)) > 2n > sigma(n) (since n is deficient, because q^k n^2 is perfect).  [See (Dris, 2017)]

LINKS

Table of n, a(n) for n=1..40.

F. J. Chen and Y.-G. Chen, On the index of an odd perfect number, Colloquium Mathematicum, 136(1) (2014), 41-49.

Jose Arnaldo Bebita Dris, On a curious biconditional involving the divisors of odd perfect numbers, Notes on Number Theory and Discrete Mathematics, 23(4) (2017), 1-13.

P. Ochem and M. Rao, Odd perfect numbers are greater than {10}^{1500}, Mathematics of Computation, 81(279) (2012), 1869-1877.

FORMULA

If N = q^k*n^2 is an odd perfect number with special prime q, then it is easy to show that sigma(n^2)/q^k = 2n^2/sigma(q^k) = gcd(n^2,sigma(n^2)).  From the last equation, it is easy to prove that D(n^2)/s(q^k) = 2s(n^2)/D(q^k) = gcd(n^2,sigma(n^2)), where D(x)=2x-sigma(x) is the deficiency of x and s(x)=sigma(x)-x is the sum of the aliquot divisors of x.

Note that, if k = 1, then sigma(q^k) < n, from which it would follow that q^k < n.  [See (Dris, 2017).]  Therefore, if k = 1, we have that N = q^k n^2 < n^3.  Using Ochem and Rao's lower bound for an odd perfect number, we get n^3 > N > 10^1500, from which we obtain n > 10^500. [See (Ochem and Rao, 2012).]

Thus, if k = 1, we have the lower bound

sigma(n^2)/q^k = 2n^2/sigma(q^k) = gcd(n^2, sigma(n^2)) > 2n > n > 10^500

which significantly improves on the corresponding result in (Chen and Chen, 2014).

The assertion k = 1 is known as the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers.

EXAMPLE

a(1) = 693 is in the sequence because

gcd((693)^2, sigma((693)^2)) = gcd(480249, sigma(480249)) > sigma(693),

where sigma(480249) = 917301 = 3*7*11^2*19^2, and 480249 = 3^4*7^2*11^2,

therefore gcd(480249, sigma(480249)) = 3*7*11^2 = 2541

but sigma(693) = 1248.

MAPLE

with(numtheory): select(n->gcd(n^2, sigma(n^2))>sigma(n), [$1..170000]); # Muniru A Asiru, Dec 06 2018

MATHEMATICA

Select[Range[10^6], GCD[#^2, DivisorSigma[1, #^2]] > DivisorSigma[1, #] &]

PROG

(GP, Sage Cell Server)

for (x=1, 1000000, if(gcd(x^2, sigma(x^2))>sigma(x), print(x)))

(PARI) isok(n) = gcd(n^2, sigma(n^2)) > sigma(n); \\ Michel Marcus, Nov 29 2018

(GAP) Filtered([1..170000], n->Gcd(n^2, Sigma(n^2))>Sigma(n)); # Muniru A Asiru, Dec 06 2018

(Python)

from sympy import divisor_sigma, gcd

for n in range(1, 170000):

    if gcd(n**2, divisor_sigma(n**2))>divisor_sigma(n):

        print(n) # Stefano Spezia, Dec 07 2018

CROSSREFS

Cf. A000203 (sigma).

Sequence in context: A292833 A004240 A004241 * A129913 A031946 A049356

Adjacent sequences:  A322151 A322152 A322153 * A322155 A322156 A322157

KEYWORD

nonn

AUTHOR

Jose Arnaldo Bebita Dris, Nov 29 2018

EXTENSIONS

More terms from Michel Marcus, Nov 29 2018

STATUS

approved

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Last modified September 27 08:39 EDT 2021. Contains 347689 sequences. (Running on oeis4.)