[seqfan] Re: A mysterious sequence found accidentally by analysis of a recycled draft.

[Paul Hanna]

Hi Thomas,
      That sequence is   *(n+1)*prime(n) + 1*   for n >= 1 with a(0) = 1.

The following PARI programs have the same output for n > 0 (with
precision):
a(n) = round( log( abs( I*(1-I) * prod(k = 1, n, 1 - I^*prime(k)*
)^prime(n) )^2 )/log(2) )
a(n) = valuation( round( 2 * prod(k = 1, n, abs(1 - I^*(prime(k)%4)* )^2
)^prime(n)  ),2)
a(n) = valuation( 2 * 2^((n+1)*prime(n)), 2)

which is simply (n+1)*prime(n) + 1.
- Paul

On Sat, Apr 13, 2024 at 1:37 AM Thomas Scheuerle <
Thomas.Scheuerle at kabelbw.de> wrote:

> Dear sequence fans,
>
> Recently a draft from Dylan A. Elias got recycled because it was abandoned.
> It did not contain complete information to reproduce it numbers.
> I was curious and looked into it and its description suggested to me
> something
> which I did put into a PARI program here:
>
> a(n) = if(n=0, 1, log(abs(I*(1-I)*prod(k = 1, n,
> 1-I^prime(k))^prime(n))^2)/log(2))
>
> It gives the sequence:
>
> 1,5,10,21,36,67,92,137,172,231,320,373,482,575,646,753,902,1063,1160,1341,1492,1607,1818,1993,2226,2523,2728,2885,3104,3271,3504,4065,...
>
> This is not the same sequence the author did originally propose as I
> failed to reproduce or understand the idea properly.
>
> This sequence is for me mysterious and puzzling, could it be interesting?
> It has odd and even terms interleaved.
> A084310(n) = gcd(n,a(n)) appears to hold by observation.
> Does anyone know into which topic this sequence may belong?
>
> thank you very much
>
> Thomas
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>