A359557 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-2) + a(n-1) are factors of a(n).

1, 2, 3, 5, 4, 6, 10, 8, 12, 20, 14, 34, 18, 26, 22, 24, 46, 70, 58, 16, 74, 30, 52, 82, 134, 36, 170, 206, 94, 60, 154, 214, 92, 102, 194, 148, 114, 262, 188, 90, 278, 138, 78, 42, 120, 48, 84, 66, 150, 54, 204, 258, 462, 180, 642, 822, 366, 132, 498, 210, 354, 282, 318, 240, 186, 426, 306

Offset: 1

Author

Scott R. Shannon, Jan 05 2023

Keywords

• nonn
• look

Comments

All terms other than 3 and 5 are even. As a(114) = 510 and a(115) = 570 both contain 2 and 5 as prime factors, all subsequent terms are multiples of 10. Likewise after 1994 terms all terms contain 2, 3, 5, 7, 11 as factors, so all subsequent terms are multiples of 2*3*5*7*11 = 2310.

The terms grow rapidly in size, e.g., a(2459) = 28318290. The smallest number not to appear is 7.

a(n) = k*m such that k = A007947(a(n-2)+a(n-1)) and m >= 1 produces the smallest k*m != a(j), j < n. - Michael De Vlieger, Jan 07 2023

Examples

			a(5) = 4 as a(3) + a(4) = 3 + 5 = 8 which contains 2 as its only distinct prime factor, and 4 is the smallest unused number to contain 2 as a factor.
a(12) = 34 as a(10) + a(11) = 20 + 14 = 34 which contains 2 and 17 as distinct prime factors, and 34 is also the smallest unused number to contain 2 and 17 as factors.
a(13) = 18 as a(11) + a(12) = 14 + 34 = 48 which contains 2 and 3 as distinct prime factors, and 18 is the smallest unused number to contain 2 and 3 as factors.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing records in red, highlighting composite prime powers in gold, squarefree semiprimes in large green, 5-smooth numbers that are not p-smooth, p > 5, in small green, 7-smooth numbers that are not p-smooth, p > 7, in magenta, 11-smooth numbers that are not p-smooth, p > 11, in cyan, and 13-smooth numbers that are not p-smooth, p > 13, in blue.

Crossrefs

Cf. A359256, A027748, A064413, A352867, A007947.

Programs

Mathematica

nn = 2^7; c[] = False; q[] = 1; f[n_] := Times @@ FactorInteger[n][[All, 1]]; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; Set[{i, j, k}, {a[1], a[2], f[a[1] + a[2]]}]; Do[m = q[k]; While[c[k m], m++]; m *= k; While[c[k q[k]], q[k]++]; Set[{a[n], c[m], i, j, k}, {m, True, j, m, f[j + m]}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jan 07 2023 *)

Python

from math import prod
from sympy import factorint
from itertools import count, islice
def agen():
    i, j, aset = 1, 2, {1, 2}; yield from [i, j]
    while True:
        m = prod(factorint(i+j))
        an = next(k*m for k in count(1) if m*k not in aset)
        i, j = j, an; aset.add(an); yield an
print(list(islice(agen(), 61))) # Michael S. Branicky, Jan 16 2023