[seqfan] Re: semiprime (A001358) analogue of A181503 Slowest-growing sequence of primes where 1/(p+1) sums to 1...

Maximilian Hasler maximilian.hasler at gmail.com
Fri Oct 29 16:04:42 CEST 2010

Er,... my comment

> %C A000001 Might also be considered as the least growing sequence
> of semiprimes such that sum( 1/a(k), k=0 ... oo ) < 1

should better be ignored -- it depends of course on the interpretation
of "least growing" ; and if we disregard the first few terms, then

a(k) = min{ x >= q^k/(q-1) | A001222(x)=2 }

would satisfy the criterion for any q>1 and asymptotically grow even
more slowly.

My apologies...

Maximilian

> %I A000001
> %S A000001 4, 4, 4, 9, 21, 33, 65, 129, 259, 514, 1027, 2049, 4097,
> 8193, 16387, 32773, 65542, 131073, 262149, 524289, 1048577, 2097157,
> 4194311, 8388609, 16777219, 33554435, 67108867, 134217731, 268435457,
> 536870918, 1073741846
> %O A000001 0,1
> %N A000001 Least semiprime >= 2^n
> %F A000001 a(k) = min{ x >= 2^k | A001222(x)=2 }
> %o A000001 (PARI) a(n)={ n=2^n; while( bigomega(n) != 2, n++); n }
> %K A000001 nonn,new
> %A A000001 M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 29 2010
>
> On Fri, Oct 29, 2010 at 7:31 AM, Richard Mathar
> <mathar at strw.leidenuniv.nl> wrote:
>>
>> More terms on behalf of http://list.seqfan.eu/pipermail/seqfan/2010-October/006328.html
>>
>> %I A000001
>> %S A000001 4,6,9,10,14,15,21,22,25,26,33,34,355,16627,76723511,17218740226618333,
>> %T A000001 374886275842473712491638217368219
>> %N A000001 Smallest growing sequence of semiprimes A001358 such that sum_{i=1..n} 1/a(i) < 1 for all n.
>> %C A000001 Semiprime variant of A075442.
>> %C A000001 The first semiprime that is not in the sequence is 35, because 1/4+1/6+1/9+..+1/34+1/35 > 1.
>> %p A000001 A := proc(n) option remember; local a,psum; if n = 1 then A001358(1); else psum := add(1/procname(i),i=1..n-1) ;
>>                for a from max(procname(n-1)+1,ceil(1/(1-psum)) ) do if isA001358(a) then if psum+1/a < 1 then
>>                return a; end if; end if; end do: end if; end proc: # R. J. Mathar, Oct 29 2010
>> %K A000001 nonn,new
>> %O A000001 1,1
>> %A A000001 Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 29 2010
>>
>