A031442 -id:A031442 - cp's OEIS frontend

A031441 a(0) = 3; for n > 0, a(n) is the greatest prime factor of PreviousPrime(a(n-1))*a(n-1)-1 where PreviousPrime(prime(k))=prime(k-1).

3, 5, 7, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23, 109, 17, 11, 19, 23

Offset: 0

Yasutoshi Kohmoto

			To get a(3) we compute PreviousPrime(7)=5, 5*7-1=34, greatest prime factor of 34 is 17, so a(3)=17.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A034970, A031442.

a[0] = 3; a[n_] := a[n] = FactorInteger[ NextPrime[ a[n-1], -1]*a[n-1] - 1][[-1, 1]]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 09 2012 *)
NestList[FactorInteger[NextPrime[#,-1]*#-1][[-1,1]]&,3,70] (* Harvey P. Dale, Dec 13 2012 *)

More terms from James Sellers

A082021 a(0) = 7; for n > 0, a(n) is the greatest prime factor of PP(PP(a(n-1)))*a(n-1)+2 where PP(n) is an abbreviation for PreviousPrime(n).

Original entry on oeis.org

7, 23, 131, 47, 643, 2459, 2000807, 503347241, 82125909539251, 9617692012399, 55555555342491359799151, 1116817987709786226917069, 578610396154837, 66992050984853, 254497141, 1660738053545999, 201525986561, 25600818891233, 796725607788661087, 23547857117470471

Offset: 0

Yasutoshi Kohmoto, May 10 2003

nonn

Tyler Busby, Table of n, a(n) for n = 0..25

Cf. A082032, A031441, A031442.

NestWhileList[FactorInteger[2+#*Prime[PrimePi[ # ]-2]][[ -1,1]] &, 7, True, 8] (* T. D. Noe, Nov 15 2006 *)
NestList[FactorInteger[NextPrime[NextPrime[#,-1],-1]#+2][[-1,1]]&,7,20] (* Harvey P. Dale, Dec 26 2017 *)

Description corrected by Rick L. Shepherd, Dec 19 2004
Corrected by T. D. Noe, Nov 15 2006
More terms from Harvey P. Dale, Dec 26 2017
a(19) from Tyler Busby, Oct 12 2023

A082132 a(0) = 5; for n > 0, a(n) is the greatest prime factor of PP(a(n-1))*a(n-1)-2 where PP(n) is an abbreviation for PreviousPrime(n).

Original entry on oeis.org

5, 13, 47, 673, 1093, 4789, 15887, 6961, 7079, 1853387, 5636791, 16319158451, 46975091221, 97536826417, 9513432505744326182381, 2335222008886384800739, 7440517660385876970522347503153, 83914607657246408236765553419, 1960358081272210906656999086971746456168551

Offset: 0

Yasutoshi Kohmoto, May 10 2003

nonn

Some of the larger entries may only correspond to probable primes.

Tyler Busby, Table of n, a(n) for n = 0..22

Cf. A082021, A031441, A031442.

p=5;for(k=1,20,print1(p,",");p=precprime(p-1)*p-2;f=factor(p);s=matsize(f)[1];p=f[s,1]) \\ Rick L. Shepherd, Dec 19 2004

Edited by Rick L. Shepherd, Dec 19 2004

a(18) from Tyler Busby, Oct 22 2023

A083557 a(n) is the greatest prime factor of 3*a(n-1)+2.

Original entry on oeis.org

3, 11, 7, 23, 71, 43, 131, 79, 239, 719, 127, 383, 1151, 691, 83, 251, 151, 13, 41, 5, 17, 53, 23, 71, 43, 131, 79, 239, 719, 127, 383, 1151, 691, 83, 251, 151, 13, 41, 5, 17, 53, 23, 71, 43, 131, 79, 239, 719, 127, 383, 1151, 691, 83, 251, 151, 13, 41, 5, 17, 53, 23

Offset: 1

Yasutoshi Kohmoto, Jun 05 2003

nonn

Conjecture: if a(1)=m then the sequence becomes cyclic, for any m.

Conjecture verified up to 25000000 by Jud McCranie, Jun 11 2003

Cf. A031439, A031440, A031442, A082021, A082132, A084923.

f[n_] := Flatten[Table[ #[[1]], {1}] & /@ FactorInteger[ 3n + 2 ]][[ -1]]; NestWhileList[f, 3, UnsameQ, All]
NestList[FactorInteger[3#+2][[-1,1]]&,3,70] (* Harvey P. Dale, Feb 21 2013 *)

lista(nn) = {print1(a = 3, ", "); for (n=1, nn, a = vecmax(factor(3*a+2)[,1]); print1(a, ", "););} \\ Michel Marcus, Jul 15 2017

G.f.: x*(3 + 11*x + 7*x^2 + 23*x^3 + 71*x^4 + 43*x^5 + 131*x^6 + 79*x^7 + 239*x^8 + 719*x^9 + 127*x^10 + 383*x^11 + 1151*x^12 + 691*x^13 + 83*x^14 + 251*x^15 + 151*x^16 + 13*x^17 + 41*x^18 + 2*x^19 + 6*x^20 + 46*x^21) / (1 - x^19) (conjectured). - Colin Barker, Jul 15 2017

cp's OEIS Frontend

A031441 a(0) = 3; for n > 0, a(n) is the greatest prime factor of PreviousPrime(a(n-1))*a(n-1)-1 where PreviousPrime(prime(k))=prime(k-1).

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A082021 a(0) = 7; for n > 0, a(n) is the greatest prime factor of PP(PP(a(n-1)))*a(n-1)+2 where PP(n) is an abbreviation for PreviousPrime(n).

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A082132 a(0) = 5; for n > 0, a(n) is the greatest prime factor of PP(a(n-1))*a(n-1)-2 where PP(n) is an abbreviation for PreviousPrime(n).

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A083557 a(n) is the greatest prime factor of 3*a(n-1)+2.

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