# Numbers not Expressible as Sum of Distinct Pentagonal Numbers

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## Theorem

The positive integer which cannot be expressed as the sum of distinct pentagonal numbers are:

- $2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30,$
- $31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61, 65,$
- $66, 67, 72, 77, 80, 81, 84, 89, 94, 95, 96, 100, 101, 102, 107, 112, 116,$
- $124, 136, 137, 141, 142, 147, 159$

## Proof

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $159$