A174292 -id:A174292 - cp's OEIS frontend

A222263 Odd numbers n such that 2n/sigma(n) - 1 = 1/x for some positive integer x.

1, 3, 15, 135, 315, 585, 819, 1155, 1365, 1485, 2295, 4455, 9009, 9405, 9945, 11115, 13923, 13965, 16965, 33705, 34335, 40365, 43785, 43875, 63063, 63855, 84825, 90675, 106425, 108927, 130815, 209385, 236115, 262845, 387387, 396495, 397575, 449295, 483945

Offset: 1

M. F. Hasler, Feb 20 2013

nonn

Odd terms from A222264, see there for motivation and further links. In particular, the powers of 2 in A222264 which correspond to perfect numbers are excluded, so all n*x from this sequence are spoof perfect numbers, cf. A174292.
Here we do not exclude n if gcd(n,x) > 1 (the first such term is 1155), which would arguably be a "reasonable" additional condition to impose.
The first term with odd x is n=a(79)=9018009, x=22021, which yields Descarte's n*x=198585576189, see also A033870, A033871 and A222262.
No other term with odd x (and thus no other odd freestyle perfect number) is known as of today, to our best knowledge. See the paper by Banks et al. for some restrictions on such numbers.
One can note that when x is even, then sigma(n)/n is of the form (2k-2)/k. For instance, for n=15 we have x=4, and sigma(n)/n = 8/5 with k=5. On the other hand, when x is odd, then sigma(n)/n is of the form (2k-1)/k. For instance, for n=9018009 we have x=22021, and sigma(n)/n = 22021/11011 with k=11011. - Michel Marcus, Nov 24 2013

			For a(3)=15, we have sigma(15)=sigma(3*5)=4*6=24, thus 2*15/sigma(15) = 30/24 = 1+1/4. Here, x=4, and if 4 was assumed to be prime, then sigma'(4)=1+4=5 (*) and sigma'(15*4) = sigma(15)*sigma'(4) = 24*5 = 120 = 2*(15*4), i.e. 15*4 is a spoof perfect number.
(*) Here, sigma' is the "freestyle" sum-of-divisors function which assumes that 4 is prime, i.e., sigma'(n)=sigma(n/4^k)*(4^(k+1)-1)/3 with 4^k || n, or k=valuation(n,4)=floor(A007814(n)/2).

Laszlo L. Toth, Table of n, a(n) for n = 1..692 (terms 1..143 terms from M. F. Hasler, 144..500 from Donovan Johnson).
W. Banks, A. Guloglu, W. Nevans and F. Saidak, Descartes Numbers, in: Anatomy of Integers, 167-174, American Mathematical Society, Providence R.I., 2008.
László Tóth, On the Density of Spoof Odd Perfect Numbers, arXiv:2101.09718 [math.NT], 2021.

Cf. A000203 = sigma, the sum-of-divisors function.

forstep(n=1,9e9,2, numerator(2*n/sigma(n)-1)==1 & print1(n","))

A033871 Divisors = 3 (mod 4) of Descartes's 198585576189.

Original entry on oeis.org

3, 7, 11, 19, 39, 63, 91, 99, 143, 147, 171, 183, 231, 247, 363, 399, 427, 507, 539, 627, 671, 819, 847, 931, 1083, 1159, 1183, 1287, 1463, 1859, 1911, 2223, 2299, 2379, 2527, 3003, 3211, 3843, 3971, 4719, 4851, 5187, 5551, 6039

Offset: 1

Naohiro Nomoto

The number 198585576189 (which is the only known odd spoof-perfect number, cf. A174292) has 486 divisors, 240 of which are congruent to 3 modulo 4. - M. F. Hasler, Feb 17 2017

			198585576189 = 3^2 * 7^2 * 11^2 * 13^2 * 19^2 * 61.

M. F. Hasler, Table of n, a(n) for n = 1..240

Cf. A033870, A222262.

Select[Divisors[198585576189],Mod[#,4]==3&] (* Harvey P. Dale, Jan 07 2023 *)

lista() = {fordiv(198585576189, d, if (d % 4 == 3, print1(d, ", ")));} \\ Michel Marcus, Jul 14 2013

select(d->d%4==1, divisors(198585576189)) \\ M. F. Hasler, Feb 17 2017

Corrected by Michel Marcus, Jul 14 2013

A222262 Divisors of Descarte's 198585576189.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 19, 21, 33, 39, 49, 57, 61, 63, 77, 91, 99, 117, 121, 133, 143, 147, 169, 171, 183, 209, 231, 247, 273, 361, 363, 399, 427, 429, 441, 507, 539, 549, 627, 637, 671, 693, 741, 793, 819, 847, 931, 1001, 1083, 1089, 1159, 1183, 1197, 1281

Offset: 1

M. F. Hasler, Feb 21 2013

The number 198585576189 = 9018009*22021 is the first and only known odd term in A058007 and A174292; 9018009 = A222263(79).

M. F. Hasler, Table of n, a(n) for n = 1..486 (complete sequence).
Index entries for sequences related to divisors of numbers

Cf. A000396, A058007, A174292, A222263, A222264.

Divisors[198585576189] (* Harvey P. Dale, Feb 22 2024 *)

PARI
```
divisors(198585576189)
```

Union of A033870 and A033871.

Offset corrected by Andrew Howroyd, Jan 17 2018

A058007 Freestyle perfect numbers n = Product_{i=1,..,k} f_i^e_i where 1 < f_1 < ... < f_k, e_i > 0, such that 2n = Product_{i=1,..,k} (f_i^(e_i+1)-1)/(f_i-1).

Original entry on oeis.org

6, 28, 60, 84, 90, 120, 336, 496, 840, 924, 1008, 1080, 1260, 1320, 1440, 1680, 1980, 2016, 2160, 2184, 2520, 2772, 3024, 3420, 3600, 3780, 4680, 5040, 5940, 6048, 6552, 7440, 7560, 7800, 8128, 8190, 8280, 9240, 9828, 9900, 10080, 10530, 11088, 11400, 13680

Offset: 1

Naohiro Nomoto, Nov 13 2000

Only one odd freestyle perfect number is known: 198585576189, found by Descartes.
This sequence consists of perfect numbers A000396 and those which aren't, called spoof-perfect numbers A174292. Roughly said, a spoof-perfect number is a number that would be perfect if one or more of its composite factors were wrongly assumed to be prime, i.e., taken as a "spoof prime". - Daniel Forgues, Nov 15 2009 (slightly rephrased)
The right hand side of the second equation in the definition, 2n = ..., equals the sum of divisors sigma(n), if all of the f_i are distinct primes. If they aren't, there arise some ambiguities: See A174292 for further discussion. - M. F. Hasler, Jan 13 2013

			n = 60 = (3^1)*(4^1)*(5^1), s = 120 = [(3^2-1)*(4^2-1)*(5^2-1)]/[(3-1)*(4-1)*(5-1)]. s-n = 120-60 = n. So 60 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, B1.

OEIS wiki, Freestyle perfect numbers.

Cf. A000396, A174292.

r[s_, n_, f_] := Catch[If[n == 1, s == 1, Block[{p, e}, Do[e = 1; While[ Mod[n, p^e] == 0, r[s*(p^(e+1) - 1)/(p-1), n/p^e, p] && Throw@True; e++], {p, Select[Divisors@n, f < # &]}]]; False]];
spoofQ[n_] := r[1/2/n, n, 1] && DivisorSigma[-1, n] != 2;
perfectQ[n_] := DivisorSigma[1, n] == 2*n;
Select[Range[10^4], spoofQ[#] || perfectQ[#]&] (* Jean-François Alcover, May 16 2017, using Giovanni Resta's code for A174292 *)

a(41)-a(45) from Amiram Eldar, Dec 27 2018

cp's OEIS Frontend

A222263 Odd numbers n such that 2n/sigma(n) - 1 = 1/x for some positive integer x.

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A033871 Divisors = 3 (mod 4) of Descartes's 198585576189.

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A222262 Divisors of Descarte's 198585576189.

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A058007 Freestyle perfect numbers n = Product_{i=1,..,k} f_i^e_i where 1 < f_1 < ... < f_k, e_i > 0, such that 2n = Product_{i=1,..,k} (f_i^(e_i+1)-1)/(f_i-1).

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