Concerning A111392.

[kohmoto]

    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
>