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

[RGWv]

?Dear Jonathan,

    Wow, we must be psychic or something like "Great minds think in similar 
directions." Anyway while I was on various airliners going to Grand 
Junction, Co. Friday I worked on this sequence with pen, paper and my TI-89. 
So now that I have answered a few of the more pressing e-mails, I started to 
work on this sequence.

{4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 69, 1497, 
259465, 4852747709, 3429487924785490781, 
305153651313989042415043589313598477}

semiPrimeQ[x_] := Plus @@ Last /@ FactorInteger[x] == 2; nextSemiPrime[n_] 
:= Block[{k = n + 1}, While[! semiPrimeQ at k, k++]; k];
a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, 
nextSemiPrime[Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Array[a, 22]

    I am trying to extend it a few more terms.

Sincerely yours, Bob.


-----Original Message----- 
From: Jonathan Post
Sent: Thursday, October 28, 2010 2:27 PM
To: Sequence Fanatics Discussion list
Subject: [seqfan] semiprime (A001358) analogue of A181503 Slowest-growing 
sequence of primes where 1/(p+1) sums to 1...

For the semiprime (A001358) analogue of
A181503 Slowest-growing sequence of primes where 1/(p+1) sums to 1
without actually reaching it.

One has:

(1/4) + (1/6) + (1/ 9) + (1/10) + (1/14) + (1/15) + (1/21) + (1/22) +
(1/25) + (1/26) + (1/33) + (1/34) = 15271237/15315300 < 1

and

(1/4) + (1/6) + (1/ 9) + (1/10) + (1/14) + (1/15) + (1/21) + (1/22) +
(1/25) + (1/26) + (1/33) + (1/34) + (1/35) = 15708817/15315300 > 1

Further steps by greedy algorithm should be simple for someone with
MAPLE or Mathematica...

Next, for 1/(1+A001358(n)):

(1/5) + (1/7) + (1/ 10) + (1/11) + (1/15) + (1/16) + (1/22) + (1/23) +
(1/26) + (1/27) + (1/34) + (1/35) + (1/36) + (1/39) + (1/40) + (1/47)
= 39139707689/39734014320  < 1

and

(1/5) + (1/7) + (1/ 10) + (1/11) + (1/15) + (1/16) + (1/22) + (1/23) +
(1/26) + (1/27) + (1/34) + (1/35) + (1/36) + (1/39) + (1/40) + (1/47)
+ (1/50) = 199671939877/198670071600  > 1


I am busy and so can't take more time right now.  But this might be made 
into an
OEIS seq, if it is not already...


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