[seqfan] Re: Two-and-five conjecture

[Vladimir Shevelev]

Dear Seqfans,

Continuing to study 2-and-5 conjecture, I understand that
the first my "method of intersections" (cf. A252282,
A252283) is not effective, and I proposed to Peter another
one, based on the sequences with the definitions
1)  Smallest k such that odd part of digital sum of 2^k
equals prime(n), n>=3 and a(n)=0, if there is no such k.
2) Let k be smallest such that s(k) = odd part of digital 
sum of 2^k is multiple of prime(n), and a(n)=0, if
there is no such k.

Note that we continue to believe that k in both definitions 
exists for every n>=3. 
However, if there exists m such that 
the m-th term of sequence 2) is less than the m-th
term of sequence 1), then 2-and-5 conjecture would
be disproven. We found such m's: 25 (see A252666)
and  14 (see A252668, if to change in the definition 2^k
with 5^k), disproving 2-and-5 conjecture.
In addition, see A252670. If 22 is the maximal term?  

Best regards,
Vladimir 


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 16 December 2014 23:47
To: seqfan at list.seqfan.eu
Subject: [seqfan] Two-and-five conjecture

Dear Seqfans,

For a prime p, denote by s(p,k) the odd part of the digital sum of p^k.
Let k_n be the smallest k such that s(p,k) is divisible by prime(n), n>=3.
I and Peter Moses conjecture that, for every n>=3, k_n=k_n(p) exists
for every p, and, moreover, the equality s(p,k_n) = prime(n) holds for
every such n, if and only if p=2 or 5.
For connected sequences, see A251964 and A252280-A252283.

Best regards,
Vladimir

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