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%I #88 Feb 14 2023 20:11:30
%S 252,2772,130284,651420,219528540,257067920340,4370154645780,
%T 292800361267260,11023640801351071740,13475008472558425746927448860,
%U 5107028211099643358085503117940,1313771981231475489737485570488833367540
%N a(1) = 252, a(n) is a(n-1) multiplied by the smallest prime factor of a(n-1)+1.
%C Conjecture: this sequence contains no terms k where k+1 is prime. (All similar sequences that start with numbers less than 252 are known to contain terms k where k+1 is prime.)
%C The next few similar sequences that seem to have this property are those that begin with a(1) = 322, 622, 664, 776, and 830. - _J. Lowell_, Mar 25 2014
%C Adding 1 to the 30th term of this sequence gives a 152-digit composite number with no factors found in ECM after hundreds of curves. - _J. Lowell_, Jan 11 2022
%C The above conjecture has now been disproved: adding 1 to the 39th term of this sequence gives a prime number. - _Andrea Concaro_, Dec 31 2022
%C Calculating a(114) requires partial factorization of a(113)+1, a 1022-digit composite number. - _Tyler Busby_, Jan 21 2023
%H Tyler Busby, <a href="/A152466/b152466.txt">Table of n, a(n) for n = 1..43</a> (terms 1..30 from Jon E. Schoenfield, terms 31..39 from Andrea Concaro)
%H Tyler Busby, Conjectured <a href="/A152466/a152466.txt">Table of n, a(n) for n = 1..113</a> using elliptic-curve factorization to obtain probable smallest factors when other methods were computationally unfeasible.
%H factordb, <a href="http://factordb.com/index.php?id=1100000004282081586">Status of a(113)+1</a>.
%F Conjecture: a(n) = A238642(a(n-1)). - _J. Lowell_, Mar 25 2014 [This conjecture fails at n=40; see the above comment from _Andrea Concaro_. - _Jon E. Schoenfield_, Jan 15 2023]
%e First term is 252. Smallest prime factor of 253 is 11, so next term is 252 * 11 = 2772.
%t a = {252}; Do[AppendTo[a, a[[ -1]]*FactorInteger[a[[ -1]] + 1][[1, 1]]], {10}]; a (* _Stefan Steinerberger_, Dec 06 2008 *)
%t NestList[#*FactorInteger[#+1][[1,1]]&,252,20] (* _Harvey P. Dale_, Apr 03 2015 *)
%o (PARI) findsmallestfactor(n)=if(isprime(n), n, forprime(p=2, 1e6, if(n%p==0, return(p))); factor(n)[1, 1])
%o lista(n)={vals=Vec([252], n); for(i=2, n, vals[i]=findsmallestfactor(vals[i-1]+1)*vals[i-1]); vals} \\ _Tyler Busby_, Jan 14 2023
%Y Cf. A020639 (smallest prime factor), A238584 (ratios of consecutive terms).
%Y Cf. A238642.
%K nonn
%O 1,1
%A _J. Lowell_, Dec 05 2008
%E More terms from _Stefan Steinerberger_, Dec 06 2008
%E Extended by _Max Alekseyev_, Sep 19 2009
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