(2! + 3! + 5! + 7! + 11! + 13! + 17! + 19! + 23! + 29! + 31! + 37! + 41! + 43! + 47! + 53!) = prime * 2^4

[Jonathan Post]

I don't know if this is known and obvious.

Numbers n such that the sum of the first n factorials
of primes is a power of 2, or a prime times a power of 2.

n such that SUM[i=1..n] p(i)! is in A093641.

a(n) = 1, 2, 3, 5, 6, 16, ...

I looked at sums up through 101!

Examples:

1: 2! = 2^1.

2: 2! + 3! = 2^3.

3: 2! + 3! + 5! = 2^7.

5: 2! + 3! + 5! + 7! + 11! = 2^4 * 2495123.

6: 2! + 3! + 5! + 7! + 11! + 13! = 2^4 * 391683923.

16:  2! + 3! + 5! + 7! + 11! + 13! + 17! + 19! + 23! + 29! + 31! + 37! + 41!
+ 43! + 47! + 53!
= 4274 883284 318648 866257 897537 783116 548765 283361 416943 855991 471976
510768
= 2^4 *  267 180205 269915 554141 118596 111444 784297 830210 088558 990999
466998 531923.

Happy Hanukkah!

-- Jonathan Vos Post
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