A152466 a(1) = 252, a(n) is a(n-1) multiplied by the smallest prime factor of a(n-1)+1.

252, 2772, 130284, 651420, 219528540, 257067920340, 4370154645780, 292800361267260, 11023640801351071740, 13475008472558425746927448860, 5107028211099643358085503117940, 1313771981231475489737485570488833367540, 40726931418175740181862052685153834393740

Offset

1,1

Comments

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.)

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

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

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

Calculating a(114) requires partial factorization of a(113)+1, a 1022-digit composite number. - Tyler Busby, Jan 21 2023

Links

Tyler Busby, Table of n, a(n) for n = 1..43 (terms 1..30 from Jon E. Schoenfield, terms 31..39 from Andrea Concaro)

Tyler Busby, Conjectured Table of n, a(n) for n = 1..113 using elliptic-curve factorization to obtain probable smallest factors when other methods were computationally unfeasible.

factordb, Status of a(113)+1.

Formula

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]

Example

First term is 252. Smallest prime factor of 253 is 11, so next term is 252 * 11 = 2772.

Mathematica

a = {252}; Do[AppendTo[a, a[[ -1]]*FactorInteger[a[[ -1]] + 1][[1, 1]]], {10}]; a (* Stefan Steinerberger, Dec 06 2008 *)
NestList[#*FactorInteger[#+1][[1, 1]]&, 252, 20] (* Harvey P. Dale, Apr 03 2015 *)

Prog

(PARI) findsmallestfactor(n)=if(isprime(n), n, forprime(p=2, 1e6, if(n%p==0, return(p))); factor(n)[1, 1])
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

Crossrefs

Cf. A020639 (smallest prime factor), A238584 (ratios of consecutive terms).

Cf. A238642.

Sequence in context: A166783 A104679 A117281 * A004535 A027815 A271496

Adjacent sequences: A152463 A152464 A152465 * A152467 A152468 A152469

Keyword

nonn,changed

Author

J. Lowell, Dec 05 2008

Extensions

More terms from Stefan Steinerberger, Dec 06 2008

Extended by Max Alekseyev, Sep 19 2009

Status

approved