A202211 a(1)=2, a(2)=3, for n >= 3, a(n) = 2*(gpf(a(n-1)) + gpf(a(n-2))) + 1, where gpf(n) is the greatest prime factor of n.

2, 3, 11, 29, 81, 65, 33, 49, 37, 89, 253, 225, 57, 49, 53, 121, 129, 109, 305, 341, 185, 137, 349, 973, 977, 2233, 2013, 181, 485, 557, 1309, 1149, 801, 945, 193, 401, 1189, 885, 201, 253, 181, 409, 1181, 3181, 8725, 7061, 1313, 817, 289, 121, 57, 61, 161, 169, 73, 173, 493

Offset

1,1

Comments

The question about the boundedness of the sequence is equivalent to the question about its eventually periodicity. For example, the sequence defined by the same recurrence relation and initial values b(1)=2 and b(2)=5 is 2, 5, 15, 21, 25, 25, 21, 25, 25, ... so is periodic for n>=4 with the period {21,25,25}.

Problem. (a) Do there exist initial terms a(1) and a(2) depending on a given N for which the sequence has the least period of length >= N? (b) Do there exist initial terms a(1) and a(2) for which the sequence has no period?

Conjecture. Problem (a) is answered in affirmative, while problem (b) is answered in the negative.

This sequence is eventually periodic and therefore bounded: a(61)=a(85)=85 [sic] and a(62)=a(86)=73. [D. S. McNeil, Dec 14 2011]

Links

Michel Marcus, Table of n, a(n) for n = 1..1000

Mathematica

a[1] := 2; a[2] := 3; a[n_] := a[n] = 2(FactorInteger[a[n - 1]][[-1, 1]] + FactorInteger[a[n - 2]][[-1, 1]]) + 1; Table[a[n], {n, 50}] (* Alonso del Arte, Dec 14 2011 *)
nxt[{a_, b_}]:={b, 2(FactorInteger[a][[-1, 1]]+FactorInteger[b] [[-1, 1]])+ 1}; Transpose[NestList[nxt, {2, 3}, 120]][[1]] (* Harvey P. Dale, Dec 15 2011 *)

Prog

(PARI) lista(nn) = my(v=vector(nn)); v[1] = 2; v[2] = 3; for (n=3, nn, v[n] = 2*(vecmax(factor(v[n-1])[, 1]) + vecmax(factor(v[n-2])[, 1])) + 1; ); v; \\ Michel Marcus, Dec 23 2025

Crossrefs

Cf. A006530 (gpf).

Sequence in context: A309755 A309701 A243896 * A104081 A395282 A267902

Adjacent sequences: A202208 A202209 A202210 * A202212 A202213 A202214

Keyword

nonn

Author

Vladimir Shevelev, Dec 14 2011

Extensions

More terms from Michel Marcus, Dec 23 2025

Status

approved