Concerning A111392

[kohmoto]

    Hi, Seqfans.
    I compared the following two sequences.

         S1 :    5, 11, 37, 13, 23, 19, 23, 37, 127
         S2 :    5, 7,  11,  13, 17, 19, 23, 29,  31

         S1 is the smallest prime factor of A111392
         S2 is (n+2)-th prime

    The terms of A111392 have a property that GCD(a(n),#p_(n+1))=1.
    So, S2 is a sequence of the possible smallest primes of A111392.

    I want to know how near they are.
    Two sequences  meet four times up to a(9 ).
    Do they meet infinitly?
    Does any law exist?

    I also want to know same thing about A112404 .
    I am sure that A112404 is nearer than A111392 to S2.
    Because each A112404(n) has A111392(n)  as a divisor.

    Yasutoshi

----- Original Message ----- 
From: "Gerald McGarvey" <Gerald.McGarvey at comcast.net>
To: "kohmoto" <zbi74583 at boat.zero.ad.jp>; <seqfan at ext.jussieu.fr>
Sent: Sunday, November 13, 2005 4:50 AM
Subject: Re: Concerning A111392.


>
> 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
>
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