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

[zak seidov]

Nice, Jonathan!

Two (un-?)related sequences
(> I don't know if this is known and obvious)

n!!!! for n=0..7:
{1,1,2,3,384,2027025,10409396852733332453861621760000,2976986715131818097048377788650104436923036193800139255873213388779256550133056640625

n!!! for n=0..7:
1,1,2,6,40320,1307674368000,12413915592536072670862289047373375038521486354677760000000000,1081396758240290900504101305800329649720646107774902579144176636573226531909905153326984536526808240339776398934872029657993872907813436816097280000000000000000000000000

Note that Mathematica considers n!!!!=(n!!)!!
and n!!!=(n!!)!, hence n!!!! =< n!!!

CHAG HANUKA SAMEACH,
Zak 

--- Jonathan Post <jvospost3 at gmail.com> wrote:
> 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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