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

[Charles Greathouse]

For the second sequence I get
23971288315924846739046729567851409860963167235674327046906491360994697009878
followed by
4002003139762412666476754379681011583097005695736125352477971429286474637011259753539336110338219384319100131328645356105327460261773656239652013467482
and
136569542749850267473213806418630671098519148802862407250943338996561349771666121944837570717043669638030269612482814339980132883379418243183266309681635140054353690721338063515751739637241790415327830652437518110636230086551877625899737971055222995864638662190069758585066570820663018120438561271094

The following term is at most
136706880825115259982539443842705091783468464291671430295675943386422659594169058181440482075770537702060920760622000611512112885689065092620523670866594066846607254062548684572378570133925988891095869638457519627884304857454648091906348689643793750450491860963055333872304254699094803530215188251039326790936647157153998296243151854164728032282229395280189647684579768987768465433627263279941944046472476777894013907675229629713104163671640972996979234803324182054701014485387427791401515071184209033940171710831051015257912132716600121119440672925246514490069720038091747256362721224766490216647
but I haven't yet found factors in two smaller numbers, ...513 and ...603.

This is as far as I'll go, since further terms would be too large even
for a b-file.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

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
> (not submitted):
>
>
> %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
>
>
> %I A000002
> %S A000002 5,7,10,11,15,16,22,23,26,27,34,35,36,39,40,47,70,1498,259466,4852747704,
> %T A000002 27172017624687178982,72672016993293266604838074954037471958
> %N A000002 Smallest growing sequence of incremented semiprimes A088707 such that sum_{i=1..n} 1/a(i) < 1 for all n.
> %C A000002 Semiprime variant of A181503.
> %e A000002 1/(4+1)+1/(6+1)+1/(9+1)+1/(10+1)+1/(14+1)+... <1.
> %p A000002 isA088707 := proc(n) numtheory[bigomega](n-1) = 2 ; end:
> %p A000002 A088707 := proc(n) option remember; local a; if n = 1 then 5; else for a from procname(n-1)+1 do
>                        if isA001358(a-1) then return a; end if; end do: end if; end proc:
> %p A000002 A := proc(n) option remember; local a,psum; if n = 1 then A088707(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 isA088707(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 A000002 nonn,new
> %O A000002 1,1
> %A A000002 Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 29 2010