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Primary pseudoperfect 42
42
42 as the third primary pseudoperfect number, and as an interior member of the finite set {1, 2, 6, 42, 1806}
Half plus a third plus a seventh plus one forty-second equals one. That is 42 itself. The same 42 sits in a tiny closed list: 1, 2, 6, 42, 1806.
What it supports
A squarefree N > 1 with 1/N plus the sum of 1/p over its distinct primes equals 1. For 42 = 2·3·7: 1/2 + 1/3 + 1/7 + 1/42 = 1. Equivalently 21 + 14 + 6 + 1 = 42. Sequence 2, 6, 42, 1806, 47058, … (OEIS A054377). The first four follow 2, 2·3, 6·7, 42·43. The fifth breaks that product rule. Separately, Σ i^n from 1 to n ≡ 1 (mod n) has exactly five positive solutions: {1, 2, 6, 42, 1806}. That set is finite.
What it does not
Not Catalan C₅. Not the order-3 magic cube. Not Adams. Later giant primary pseudoperfects are not extra 42s.
Calculation
1/2 + 1/3 + 1/7 + 1/42 = 1. {1, 2, 6, 42, 1806} solves Σ_{i=1}^{n} i^n ≡ 1 (mod n).
Primary pseudoperfect identity, then the finite power-sum congruence.
ExactnessExact
Form42
Unitinteger
Strengthhard