In mathematics a superperfect number is a positive integer n that satisfies

where σ is the divisor function. Superperfect numbers are a generalization of perfect numbers. The term was coined by Suryanarayana (1969).^[1]

The first few superperfect numbers are

2, 4, 16, 64, 4096, 65536, 262144 (sequence A019279 in OEIS).

If n is an even superperfect number then n must be a power of 2, 2^k, such that 2^k+1-1 is a Mersenne prime.^[1][2]

It is not known whether there are any odd superperfect numbers. An odd superperfect number n would have to be a square number such that either n or σ(n) is divisible by at least three distinct primes.^[2] There are no odd superperfect numbers below 7x10²⁴.^[1]

Generalisations [edit]

Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy

corresponding to m=1 and 2 respectively. For m ≥ 3 there are no even m-superperfect numbers.^[1]

The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy^[3]

With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect.^[4] Examples of classes of (m,k)-perfect numbers are:

m	k	(m,k)-perfect numbers	OEIS sequence
2	3	8, 21, 512	A019281
2	4	15, 1023, 29127	A019282
2	6	42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024	A019283
2	7	24, 1536, 47360, 343976	A019284
2	8	60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072	A019285
2	9	168, 10752, 331520, 691200, 1556480, 1612800, 106151936	A019286
2	10	480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296	A019287
2	11	4404480, 57669920, 238608384	A019288
2	12	2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120	A019289
3	any	12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, ...	A019292
4	any	2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, ...	A019293

Notes [edit]

References [edit]

v t e Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number Factorization forms Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number Constrained divisor sums Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hemiperfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Practical number With many divisors Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number Aliquot sequence-related Untouchable number Amicable number Sociable number Betrothed number Other sets Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number