Concerning A111392.

[Gerald McGarvey]

Very nice sequence. I ran the following PARI/GP code to get some terms
after the first term and their factors...

t=10;
for(n=2,t,print(prod(i=1,n-1,prod(k=1,i,prime(k))+prod(k=i+1,n,prime(k)))));

5
187
162319
10697595389
63619487169453143
74365399061678006800073593
11864736003419293844093922527852416537
601642845102734414280661105098046392912578705726003
18053982712642869556235207711881953337147219875860418947153470003097

for(n=2,t,print(factor(prod(i=1,n-1,prod(k=1,i,prime(k))+prod(k=i+1,n,prime(k)))))); 


5
11 * 17
37 * 41 * 107
13^2 * 17^2 * 23 * 89 * 107
23 * 101 * 353 * 1031 * 5011 * 15017
19 * 47 * 139 * 2531 * 5431 * 17047 * 30047 * 85091
23 * 53 * 113 * 127 * 167 * 239 * 257 * 283 * 6101 * 46399 * 510529 * 1616621
37 * 47 * 263 * 599 * 797 * 853 * 3821 * 9221 * 12097 * 98887 * 1062557 * 
7436459 * 9699713
127 * 283 * 311 * 443 * 751 * 1181 * 1231 * 1427 * 9013 * 17239 * 41023 * 
62549 * 245471 * 762037 * 9700357 * 3234846617

(the output is edited to be simpler)

Regards,
Gerald

At 10:17 PM 11/11/2005, kohmoto wrote:
>    Hi, Robert
>
>    Thanks for editing my sequence.
>    I think your calculation is correct.
>
>    I did (2+3*5)+(2*3+5) instead of (2+3*5)*(2*3+5) at a(3). Naturally, 
> 187 is correct.
>    a(1) is depend on the definition.
>    I think both 1 and 2 are OK.
>
>    Yasutoshi
>
>----- Original Message ----- From: "Robert G. Wilson v" <rgwv at rgwv.com>
>To: "Yasutoshi Kohmoto" <zbi74583 at boat.zero.ad.jp>
>Sent: Saturday, November 12, 2005 6:20 AM
>Subject: Concerning A111392.
>
>
>>Dear Sir,
>>
>>What a wonderful and surprising unique sequence. However I am having a
>>difficult time extending your sequence by your definition. I hope that I have
>>implemented it correctly. If so I get the following terms:
>>1, 5, 187, 162319, 10697595389, 63619487169453143, 
>>74365399061678006800073593,
>>..., which obviously differ from yours. Can you please advise?
>>
>>Sincerely yours,
>>
>>Robert G. 'Bob' Wilson, V
>>
>>
>>
>>%I A111392
>>%S A111392 2,5,28,162319
>>%N A111392 a(n) = Product_{1<=i<n} (Product_{1<=k<=i} p_k + 
>>Product_{i<k<=n} p_k).
>>%C A111392 This is a "Proof of existence of infinite primes" sequence. 
>>Proof. Let N = Product_{1<=i<n} (Product_{1<=k<=i} p_k + Product_{i<k<=n} 
>>p_k). Suppose there are only a finite number of primes p_i, 1<=i<=n. If N 
>>is prime, then for all i, not (N=p_i). Because, for all i, p_i<N. If N is 
>>composite, then it must have a prime divisor p which is different from 
>>primes p_i. Because, for all i, not (N_1=0, Mod p_i).
>>%t A111392 f[n_] := Product[ (Product[Prime[k], {k, i}] + Product[ 
>>Prime[k], {k, i + 1, n}]), {i, n - 1}]
>>%Y A111392 Cf. A024451.
>>%K A111392 nonn
>>%O A111392 1,1
>>%A A111392 Yasutsohi Kohmoto zbi74583(AT)boat.zero.ad,jp