A055153 -id:A055153 - cp's OEIS frontend

A364976 3-abundant numbers k such that k/(sigma(k)-3*k) is an integer.

180, 240, 360, 420, 540, 600, 780, 1080, 1344, 1872, 1890, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3744, 4284, 4320, 4680, 5292, 5376, 5796, 6048, 6552, 7128, 7440, 8190, 10416, 13776, 14850, 18600, 19824, 19872, 20496, 21528, 22932, 25056, 26208, 26496, 26784

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

Analogous to A153501 as 3-abundant numbers (A068403) are analogous to abundant numbers (A005101).

Numbers k such that the sum of the divisors of k except for one of them is equal to 3*k.

			180 is a term since sigma(180) - 3*180 = 6 > 0 and 180 is divisible by 6.

Amiram Eldar, Table of n, a(n) for n = 1..576 (terms below 2.5*10^10)

Subsequence of A068403.
A027687 is a subsequence.
Cf. A000203 (sigma), A005101, A005820, A055153, A153501, A160320, A218404, A218406, A329189, A364977.

Select[Range[27000], (d = DivisorSigma[1, #] - 3*#) > 0 && Divisible[#, d] &]

is(n) = {my(d = sigma(n) - 3*n); d > 0 && n%d == 0;}

A306373 Integers m such that the sum of the first k divisors is equal to 2*m for some k less than the number of divisors of m.

Original entry on oeis.org

120, 672, 4320, 4680, 26208, 523776, 20427264, 29795040, 34369920, 96445440, 197064960, 459818240, 557107200

Offset: 1

Michel Marcus, Feb 11 2019

3-perfect numbers (A005820) are terms.
All known terms of A055153 (abundancy 7/2) are terms.
1907020800 (with abundancy 23/6) is a term too.
A055153 is a subsequence, because no term of that sequence may be odd and so for each k in A055153 we have 2*k = sigma(k) - k - k/2. - Charlie Neder, Feb 12 2019

Cf. A005820 (3-perfect numbers), A055153 (abundancy 7/2).

Cf. A064510, A194472 (both with equal to m rather than to 2*m).

isok(n) = {if (sigma(n) < 2*n, return (0)); my(d = divisors(n), s = 0); for (k=1, #d-1, s += d[k]; if (s == 2*n, return (1)); if (s > 2*n, break);); return (0);}

is(n) = my(d = divisors(n), s = vecsum(d) - d[#d]); forstep(i = #d-1, 1, -1, if(s <= 2*n, return(s == 2*n)); s-=d[i]); 0 \\ David A. Corneth, Feb 11 2019

a(11)-a(13) from Jinyuan Wang, Feb 11 2019

A227882 Known number of n_multiperfect numbers that can produce an hemiperfect of abundancy (2*n-1)/2.

Original entry on oeis.org

1, 3, 19, 0, 87, 117, 0, 30, 0, 0

Offset: 2

Michel Marcus, Oct 25 2013

The hemiperfect that are obtained are coprime to p = 2*n-1.

When p=2*n-1 is prime, if m is a n-multiperfect is such that valuation(m, p) = 1, then let's define k = m/p, sigma(k) = sigma(m/p) = sigma(m)/sigma(p) = (n*m)/(p+1) = (n*m)/(2*n) = m/2. So sigma(k)/k = m/(2*k) = (k*p)/(2*k) = p/2 = (2*n-1)/2.

			a(2) = 1, since the only perfect number multiple of 3 is 6, and 6/3=2 has abundancy 3/2.
a(3) = 3, since the 3 known hemiperfect of abundancy 5/2 are coprime to 5.
a(5) = a(8) = a(11) = 0, since for those n, 2*n-1 is not prime.
a(10) is also 0, since all known 10-multiperfect are at least divisible by 19^2.

Achim Flammenkamp, The Multiply Perfect Numbers Page
G. P. Michon, Multiperfect and hemiperfect numbers

Cf. A000396 (2), A005820 (3), A027687 (4), A046060 (5), A046061 (6), A007691 (integer abundancy).
Cf. A141643 (5/2), A055153 (7/2), A141645 (9/2), A159271 (11/2), A160678 (13/2), A159907 (half-integer abundancy).
Cf. A006254.

cp's OEIS Frontend

A364976 3-abundant numbers k such that k/(sigma(k)-3*k) is an integer.

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A306373 Integers m such that the sum of the first k divisors is equal to 2*m for some k less than the number of divisors of m.

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A227882 Known number of n_multiperfect numbers that can produce an hemiperfect of abundancy (2*n-1)/2.

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