Perfect, abundant, deficient, amicable, sociable, weird, ... numbers

[lpetrich]

For positive integers n, one can construct a divisor-sum function σ:

\( n = \prod_i (p_i)^{m_i} ;\ \sigma(n) = \prod_i \frac{(p_i)^{m_i} - 1}{p_i - 1} \)

More generally, σ_k is the sum of powers k of divisors, with σ₀ being the number of divisors.

\( \sigma_k(n) = \prod_i \frac{(p_i)^{k m_i} - 1}{(p_i)^k - 1} ;\ \sigma_0(n) = \prod_i m_i \)

A  Perfect number is a number n which is the sum of all its proper divisors. That is, σ = 2n, since the sum in σ includes n itself. One may define a restricted divisor sum s = σ - n that omits n. Thus for a perfect number, s = n.

All even perfect numbers are known. They have form 2^n-1 * (2ⁿ - 1) where (2ⁿ - 1) is a prime number, a  Mersenne prime In fact, every Mersenne prime has an associated perfect number. For (2ⁿ - 1) to be a prime number, n must also be a prime number, though only some prime numbers give Mersenne primes. The smallest one that doesn't is 11, and 2^11-1 = 8191 = 23*89. The previous four, for 2, 3, 5, 7, were known in antiquity: 3, 7, 31, 127, along with their perfect numbers: 6, 28, 496, 8128.

If you want to help search for large Mersenne primes, consider participating in the  Great Internet Mersenne Prime Search


It is not known whether or not odd perfect numbers exist, and some strong constraints have been placed on them. They must be greater than 10¹⁵⁰⁰, for instance.

 

[lpetrich]

Every perfect number is a  Harmonic divisor number - a number's divisors have a harmonic mean that is an integer. The harmonic mean is the reciprocal of the average of reciprocals, and HDN = n*σ₀/σ₁

Some HDN's are not perfect numbers, however, and A001599 - OEIS has
1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720, ...

As with odd perfect numbers, there are strong constraints on what nontrivial odd HDN's might exist, like being greater than 10²⁴.


 Superperfect number

Found by repeating the divisor function: σ⁽²⁾ = σ(σ) = 2n

A019279 - OEIS
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, ...

All even ones are known: 2ⁿ such that 2ⁿ⁺¹-1 is a Mersenne prime.

No odd ones are known, though the smallest one is known to be greater than 7*10²⁴.

For σ^(m) = 2n with m >= 2, no even ones are known.

 

[lpetrich]

 Multiply perfect number have σ = k*n for some integer k, making k-perfect numbers. With the restricted divisor function s, s = (k-1)*n

For k = 1, the only solution is n = 1.

For k = 2, we have the familiar perfect numbers.

K-perfect numbers get very large very quickly. The smallest 3-perfect number is 120, and the number increases as exp(exp(0.9976*k / e^gamma)) where gamma is the Euler-Mascheroni constant(?).

One can also define (m,k)-superperfect numbers: σ^(m) = k*n

A019279 - OEIS - (2,2)-perfect numbers: 2^n-1 if 2ⁿ-1 is a Mersenne prime (all even ones)

A019281 - OEIS - (2,3)-perfect numbers: 8, 21, 512 with the next one >8*10¹² if it exists

A019282 - OEIS - (2,4)-perfect numbers: 15, 1023, 29127, 355744082763 with the next one >4*10^12 if it exists

A019283 - OEIS - (2,6)-perfect numbers: 21*2^n-1 if 2ⁿ-1 is a Mersenne prime, with additional numbers 160, 550095 with the next one >8*10¹¹ if it exists

A019284 - OEIS - (2,7)-perfect numbers: 24, 1536, 47360, 343976, 572941926400 with the next one >4*10¹² if it exists

A019285 - OEIS - (2,8)-perfect numbers: 15*2^n-1 if 2ⁿ-1 is a Mersenne prime > 3, with additional numbers 4092, 16368, 58254, 65472, 116508, 466032, 710400, 1864128, 4190208, 67043328, 119304192, 268173312, 1908867072, 7635468288, 711488165526, 1098437885952, 1422976331052 with the next one >5*10¹¹ if it exists

A019286 - OEIS - (2,9)-perfect numbers: 168, 10752, 331520, 691200, 1556480, 1612800, 106151936, 5099962368, 4010593484800 with the next one >5*10¹¹ if it exists

A019287 - OEIS - (2,10)-perfect numbers: 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296, 14763499520, 38385098752 with the next one >4*10¹² if it exists

A019288 - OEIS - (2,11)-perfect numbers: 4404480, 57669920, 238608384, 53283599155200, 2914255525994496, 3887055949004800, ...

A019289 - OEIS - (2,12)-perfect numbers: 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120, 16785793024, 22648550400, 36051025920, 51001180160, 144204103680, 6640556211576, 82863343951872, 182140970374656, 480965999895576, 590660008673280, 886341160140800, 5562693163417600, 9386507580211200, ...

 

[lpetrich]

A019292 - OEIS - (3,k): 1, 12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, 6882, 7616, 9114, 14592, 18288, 22848, 32704, 40880, 52416, 53760, 54864, 56448, 60960, 65472, 94860, 120960, 122640, 169164, 185535, 186368, 194432, 196137, 201872, 208026, 286160

The k's for them: 1, 10, 12, 20, 5, 6, 10, 16, 21, 24, 20, 34, 28, 30, 36, 14, 16, 15, 24, 21, 24, 20, 10, 26, 20, 34, 20, 30, 27, 24, 48, 48, 36, 19, 32, 35, 39, 16, 18, 38, 24

A019293 - OEIS - (4,k): 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, 336, 455, 512, 672, 896, 960, 992, 1023, 1280, 1536, 1848, 2040, 2688, 4092, 5472, 5920, 7808, 7936, 10416, 11934, 16352, 16380, 18720, 20384, 21824, 23424, 24564, 29127, 30240, 33792, 36720, 41440

The k's for them: 1, 4, 5, 6, 20, 21, 12, 30, 32, 20, 10, 63, 28, 18, 30, 40, 80, 24, 28, 60, 62, 48, 36, 65, 48, 21, 124, 57, 78, 84, 32, 30, 63, 110, 80, 84, 80, 125, 48, 24, 35, 52, 155, 48, 156, 168, 96, 66, 78, 117, 32, 192, 93, 208, 96

 

[lpetrich]

 Hyperperfect number - numbers n that satisfy n = 1 + k(σ − n − 1) for some k, thus omitting all trivial divisors: 1 and n.

A034897 - OEIS - hyperperfect numbers
6, 21, 28, 301, 325, 496, 697, 1333, 1909, 2041, 2133, 3901, 8128, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833

A034898 - OEIS - their corresponding k values
1, 2, 1, 6, 3, 1, 12, 18, 18, 12, 2, 30, 1, 11, 6, 2, 60, 48, 19, 132, 132, 10, 192, 2, 31, 168, 108, 66, 35, 252, 78, 132, 342, 366, 390, 168, 348, 282, 498, 540, 546, 59, 12, 378, 438, 4, 222, 336, 18, 660, 138, 798, 810, 528, 450, 75, 252, 150, 948, 162

A000396 - OEIS - perfect numbers
6, 28, 496, 8128, 33550336, 8589869056, 137438691328

A007592 - OEIS - hyperperfect numbers that are not perfect
21, 301, 325, 697, 1333, 1909, 2041, 2133, 3901, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053

 

[lpetrich]

For perfect numbers, σ = 2n or s = n. If the number is greater or less, what do we get?

 Abundant number n has σ > 2n
A005101 - OEIS
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270

 Deficient number n has σ < 2n
A005100 - OEIS
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86

The smallest odd abundant number is 945.

Every multiple of an abundant number or a perfect number is an abundant number. Every abundant number that is not is a  Primitive abundant number
A071395 - OEIS
20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030, 4070, 4095, 4216, 4288

An abundant number is a  Superabundant number if σ/n > σ(m)/m for all m < n -- it is more abundant than any previous number.
A004394 - OEIS
1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600

Their prime exponents are non-increasing for increasing prime factors.

Turning to deficient numbers, all prime numbers and their powers are deficient.

All proper divisors of deficient numbers and perfect numbers are deficient.

 

[lpetrich]

A  Semiperfect number is the sum of at least some of its proper divisors.
A005835 - OEIS
6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264

Every multiple of a semiperfect number is semiperfect. A semiperfect number that is not a multiple of some other one is primitive.
A006036 - OEIS
6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, 650, 748, 770, 910, 945, 1184, 1190, 1312, 1330, 1376, 1430, 1504, 1575, 1610, 1696, 1870, 1888, 1952, 2002, 2030, 2090, 2170, 2205, 2210, 2470, 2530, 2584, 2590, 2870, 2990, 3010, 3128, 3190, 3230, 3290, 3410, 3465, 3496, 3710, 3770, 3944, 4070, 4095, 4130, 4216, 4270, 4288, 4408, 4510, 4544, 4672, 4690, 4712, 4730, 4970


A  Weird number is abundant but not semiperfect.
A006037 - OEIS
70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670

If an odd weird numbers exists, then it must be greater than 10²¹.

If n is weird and p a prime greater than σ, then p*n is also weird. This leads to identification of the "primitive weird numbers".
A002975 - OEIS
70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448

 

[lpetrich]

 Untouchable number - a positive integer that is not the sum of proper divisors of any other number.
A005114 - OEIS
2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658

The only known odd one is 5.

There are infinitely many untouchable numbers.

According to Goldbach's conjecture, every even number greater than 2 can be written as the sum of two prime numbers. 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, 10 = 5 + 5 = 7 + 3, 12 = 7 + 5, 14 = 7 + 7 = 11 + 3, 16 = 13 + 3 = 11 + 5

A stricter version of it says that these two primes must be distinct. Of this list, that rules out 4 and 6. For a product of distinct Goldbach-conjecture primes p*q, s(p*q) = p + q + 1. So if this version of this conjecture is true, then 5 is the only odd untouchable number.

 

[lpetrich]

 Amicable numbers

These are pairs of numbers which are sums of each others' proper divisors. Perfect numbers are a degenerate case, with both equal.

A259180 - OEIS - Amicable pairs, in sequence
A002025 - OEIS - Smaller member of each amicable pair
A002046 - OEIS - Larger member of each amicable pair

220, 284 / 1184, 1210 / 2620, 2924 / 5020, 5564 / 6232, 6368 / 10744, 10856 / 12285, 14595 / 17296, 18416 / 63020, 76084 / 66928, 66992 / 67095, 71145 / 69615, 87633 / 79750, 88730 / 100485, 124155 / 122265, 139815 / 122368, 123152 / 141664, 153176 / 142310, 168730 / 171856, 176336 / 176272, 180848 / 185368, 203432 / 196724, 202444 / 280540, 365084

There are some formulas for generating amicable numbers, but they are not comprehensive.

Thabit ibn Qurra:
p = 3*2^n-1 - 1
q = 3*2ⁿ - 1
r = 9*2^2n-1 - 1
are all primes, with n > 1, then 2ⁿ*p*q and 2ⁿ*r are a pair of amicable numbers. It gives n = 2: (220, 284), n = 4: (17296, 18416), n = 7: (9363584, 9437056) with no others known.

Euler:
p = (2^n-m + 1)*2^m - 1
q = (2^n-m + 1)*2ⁿ - 1
r = (2^n-m + 1)²*2^m+n - 1
are all primes, with n > m > 0. Thabit's formula is for m = n - 1. Euler has some additional ones: (m,n) = (1,8), (29,40) with no others being known.


Regular amicable pairs: Let m = g*M, n = g*N, where g = gcd(m,n), meaning that M and N are coprime. If both M and N are coprime to g and square free, then the pair is regular. For instance, (220, 284) is regular. 220 = 4*55, 284 = 4*71.

 

[lpetrich]

 Sociable number - each list of them forms a sequence where each one is the sum of proper divisors of the previous one, looping around to the beginning. They thus generalized perfect numbers and amicable numbers.

The smallest-number 4-sequence: 1264460, 1547860, 1727636, 1305184

The only known 5-sequence: 12496, 14288, 15472, 14536, 14264

The only known 28-sequence: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716

They are known for sequence lengths 1 (perfect numbers), 2 (amicable numbers), 4, 5, 6, 8, 9, and 28, and someone has conjectured that there are no sequences with lengths 3 mod 4: 3, 7, 11, ...

 

[lpetrich]

Sociable numbers may be interpreted as the limit cycles of the sum of proper divisors, s for number n. Perfect numbers are thus fixed points, one-member limit cycles, and amicable numbers two-member limit cycles.

Repeating the proper-divisor sum function will produce either infinite sequences or limit cycles.

I tried a numerical experiment on this issue, checking on starting numbers from 1 to 10,000. I used a cutoff at 100 iterations because I have no way to test for running off to infinity.

Of these, 8720 went to zero, with 1229 taking two iterations. Those were the prime numbers. One of them took only one iteration: 1. Of the rest, 1045 took three iterations, with smaller numbers for more iterations. There were two that took 77 iterations, and there were two that took 88 iterations. That leads to the possibility that some of the cut off ones may eventually end up in zero, but I'd have to port my code from Mathematica to C++ to be sure.

There were 1056 that ran over my iteration limit. The first 10 numbers to do so were 138, 150, 168, 222, 234, 276, 306, 312, 396, 528.

There were 224 that ended up in some sociable number. Of these, 93 ended up in a perfect number, 78 in 6, 1 in 28, 12 in 496, and 2 in 8128. It usually took around 4 or 5 iterations to converge, though that was less for relatively small numbers. Of the 131 that remained, 121 converged to amicable numbers and 10 to longer-cycle sociable numblers, like the 5-cycle ones.