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Problem 333

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Special partitions

All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as 2ⁱ×3^j, where i,j ≥ 0.

Let’s consider only those such partitions where none of the terms can divide any of the other terms.
For example, the partition of 17 = 2 + 6 + 9 = (2¹×3⁰ + 2¹×3¹ + 2⁰×3²) would not be valid since 2 can divide 6. Neither would the partition 17 = 16 + 1 = (2⁴×3⁰ + 2⁰×3⁰) since 1 can divide 16. The only valid partition of 17 would be 8 + 9 = (2³×3⁰ + 2⁰×3²).

Many integers have more than one valid partition, the first being 11 having the following two partitions.
11 = 2 + 9 = (2¹×3⁰ + 2⁰×3²)
11 = 8 + 3 = (2³×3⁰ + 2⁰×3¹)

Let’s define P(n) as the number of valid partitions of n. For example, P(11) = 2.

Let’s consider only the prime integers q which would have a single valid partition such as P(17).

The sum of the primes q <100 such that P(q)=1 equals 233.

Find the sum of the primes q <1000000 such that P(q)=1.

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特殊分划

所有的正整数可以进行如下的分划；分划的每一项都能表示成2ⁱ×3^j，其中i,j ≥ 0。

进一步地，我们只考虑任意一项不能整除其它项的分划。
例如，分划17 = 2 + 6 + 9 = (2¹×3⁰ + 2¹×3¹ + 2⁰×3²)是不合理的，因为2可以整除6。分划17 = 16 + 1 = (2⁴×3⁰ + 2⁰×3⁰)同样是不合理的，因为1可以整除16。对17唯一合理的分划为8 + 9 = (2³×3⁰ + 2⁰×3²)。

许多整数都有不止一种合理的分划，其中最小的为11，有如下两种分划。
11 = 2 + 9 = (2¹×3⁰ + 2⁰×3²)
11 = 8 + 3 = (2³×3⁰ + 2⁰×3¹)

我们记P(n)为对n合理的分划数目。例如，P(11) = 2。

再进一步，我们只考虑只有一种合理分划的素数q，如P(17)=1。

所有使得P(q)=1的素数q <100之和是233。

求所有使得P(q)=1的素数q <1000000之和。

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