A359874 a(1) = 1, a(2) = 2. For n > 2, a(n) is the least novel power of the greatest prime divisor of a(n-2) + a(n-1).

1, 2, 3, 5, 4, 9, 13, 11, 27, 19, 23, 7, 25, 8, 121, 43, 41, 49, 125, 29, 1331, 17, 337, 59, 14641, 343, 1873, 277, 1849, 1063, 169, 161051, 2687, 81869, 21139, 37, 2647, 61, 677, 1681, 131, 151, 47, 1771561, 761, 295387, 1369, 74189, 257, 37223, 937, 53, 19487171

Offset

1,2

Comments

In other words, a(n) is the least novel number p^k, where p is the greatest prime divisor of a(n-2) + a(n-1) and k >= 1. All terms are prime powers, though it is not known if every such number appears. Primes to the first power do not occur in natural order (e.g., 43 precedes 29).

Links

Table of n, a(n) for n=1..53.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing and labeling primes in red.

Example

a(3) = 3 because a(1) + a(2) = 3, and no power of 3 has yet appeared.
a(4) = 5 because a(2) + a(3) = 5, and no power of 5 has yet appeared.
a(15) = 121 because a(13) + a(14) = 33 which has greatest prime divisor 11. But 11^1 has already occurred at a(8), so the least novel power of 11 is 11^2 = 121.

Mathematica

nn = 120; c[_] = False; q[_] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = FactorInteger[a[1] + a[2]][[-1, 1]]; u = 2; Do[k = q[s]; While[c[s^k], k++]; While[c[s^q[s]], q[s]++]; k = s^k; Set[{a[n], c[k], s}, {k, True, FactorInteger[a[n - 1] + k][[-1, 1]]}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jan 16 2023 *)

Crossrefs

Cf. A000961.

Sequence in context: A332301 A360281 A069202 * A244984 A373998 A247225

Adjacent sequences: A359871 A359872 A359873 * A359875 A359876 A359877

Keyword

nonn

Author

David James Sycamore, Jan 16 2023

Extensions

More terms from Michael De Vlieger, Jan 16 2023

Status

approved