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Category Number Generators

Number of almost perfect numbers to generate Range: 1 to 53 (limited by JS integer precision)

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#	n	k (if 2^k)	σ(n)	2n − 1	Divisors

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About

An almost perfect number is an integer n for which the sum-of-divisors function σ(n) = 2n − 1. Equivalently, the sum of proper divisors falls exactly 1 short of n itself. Every known almost perfect number is a power of 2. Whether a non-power-of-2 almost perfect number exists remains an open problem in number theory. Miscounting divisors or confusing "almost perfect" with "deficient" leads to classification errors in combinatorial and cryptographic applications that rely on divisor function properties.

This tool operates in two modes. The generator mode computes the first N almost perfect numbers directly as 2^k values. The range search mode performs brute-force divisor summation across an arbitrary interval [a, b] to independently verify each candidate. For ranges exceeding 100,000 integers, computation is offloaded to a Web Worker. Note: the tool approximates large σ(n) calculations using trial division up to √n. Results for n beyond 2⁵³ lose precision due to JavaScript floating-point limits.

Formulas

A positive integer n is almost perfect if the sum-of-divisors function satisfies:

σ(n) = 2n − 1

The sum-of-divisors function is computed as:

σ(n) = ∑d | n d

This equals the sum over all positive divisors d of n. The deficiency of n is defined as:

D(n) = 2n − σ(n)

An almost perfect number has deficiency exactly 1. For powers of 2, the divisor sum follows a geometric series:

σ(2^k) = k∑i=0 2ⁱ = 2^k+1 − 1

Since 2^k+1 − 1 = 2 ⋅ 2^k − 1, every power of 2 satisfies the condition. The brute-force verification algorithm computes σ(n) by trial division up to √n, yielding O(√n) per candidate.

Where: σ(n) = sum of all positive divisors of n; D(n) = deficiency; k = non-negative integer exponent; d | n means d divides n.

Reference Data

k	n = 2^k	σ(n)	2n − 1	Proper Divisor Sum	Deficiency
0	1	1	1	0	1
1	2	3	3	1	1
2	4	7	7	3	1
3	8	15	15	7	1
4	16	31	31	15	1
5	32	63	63	31	1
6	64	127	127	63	1
7	128	255	255	127	1
8	256	511	511	255	1
9	512	1023	1023	511	1
10	1024	2047	2047	1023	1
11	2048	4095	4095	2047	1
12	4096	8191	8191	4095	1
13	8192	16383	16383	8191	1
14	16384	32767	32767	16383	1
15	32768	65535	65535	32767	1
16	65536	131071	131071	65535	1
17	131072	262143	262143	131071	1
18	262144	524287	524287	262143	1
19	524288	1048575	1048575	524287	1
20	1048576	2097151	2097151	1048575	1

Frequently Asked Questions

For n = 2^k, the divisors are {1, 2, 4, ..., 2^k}. Their sum is a geometric series equaling 2^(k+1) − 1 = 2n − 1, which is exactly the almost perfect condition. Whether any odd almost perfect number exists is an unsolved problem. Research by Kishore (1978) showed that any odd almost perfect number must exceed 10^30 and have at least 6 distinct prime factors.

A perfect number has σ(n) = 2n (deficiency 0), meaning the proper divisors sum exactly to n. A deficient number has σ(n) < 2n (deficiency ≥ 1). An almost perfect number is the specific case where deficiency equals exactly 1. All almost perfect numbers are deficient, but not all deficient numbers are almost perfect. For example, 9 is deficient (σ(9) = 13, deficiency = 5) but not almost perfect.

Theoretically yes. The range search computes σ(n) by trial division for every integer in the interval and checks if σ(n) = 2n − 1. If such a number existed within the range, it would be flagged. In practice, computational searches up to 10^18 have found none. The tool supports ranges up to 10^7 comfortably in the Web Worker; larger ranges will be slower due to O(√n) per candidate.

JavaScript uses IEEE 754 double-precision floats, which represent integers exactly up to 2^53 (9,007,199,254,740,992). The generator mode correctly produces 2^k for k up to 52. For k = 53 and beyond, the generated values lose precision. The range search mode is constrained to Number.MAX_SAFE_INTEGER. Inputs exceeding this limit trigger a validation error toast.

The divisors of 2^k are exactly {2^0, 2^1, ..., 2^k}. By the geometric series formula, their sum equals (2^(k+1) − 1) / (2 − 1) = 2^(k+1) − 1. Since 2 × 2^k = 2^(k+1), we get σ(2^k) = 2 × 2^k − 1 = 2n − 1. This algebraic identity confirms every power of 2 is almost perfect without needing to enumerate divisors.

Each candidate n requires O(√n) operations for trial division. For a range [a, b], total work is approximately Σ√n for n in [a, b]. A range of 1,000,000 integers near 10^6 involves roughly 10^9 operations total, completing in 2-5 seconds in a Web Worker. Ranges exceeding 10^7 integers may take 30+ seconds. A progress bar and cancel button are provided for long computations.