A348872 After a(1) = 1, the sequence is always extended with the smallest divisor d (not yet present in the sequence) of the last term t. If d doesn't exist, we extend the sequence with 2*t + 1 and repeat. See the Comments section for more details.

1, 3, 7, 15, 5, 11, 23, 47, 95, 19, 39, 13, 27, 9, 19, 39, 79, 159, 53, 107, 215, 43, 87, 29, 59, 119, 17, 35, 71, 143, 287, 41, 83, 167, 335, 67, 135, 45, 91, 183, 61, 123, 247, 495, 33, 67, 135, 271, 543, 181, 363, 121, 243, 81, 163, 327, 109, 219, 73, 147, 21, 43, 87, 175, 25, 51, 103, 207, 69, 139, 279, 31, 63

Offset

1,2

Links

Table of n, a(n) for n=1..73.

Michael De Vlieger, Scatterplot of a(n), for n=1..2^16, accentuating smaller terms for clarity, labeling a(n) for n=1..32.

Example

a(1) = 1 by definition; as 1 has no available divisor yet present in the sequence, we produce a(2) = 2*1 + 1 = 3.
a(2) = 3; as 3 has no available divisor yet present in the sequence, we produce a(3) = 2*3 + 1 = 7;
a(3) = 7; 7 has no available divisor yet present in the sequence, we produce a(4) = 2*7 + 1 = 15;
a(4) = 15; as 15 has 5 as its smallest divisor not yet present in the sequence, we have a(5) = 5.

Mathematica

a[1]=1; a[n_]:=a[n]=If[(s=Complement[Rest@Divisors@a[n-1], Array[a, n-1]])!={}, Min@s, 2a[n-1]+1]; Array[a, 73] (* Giorgos Kalogeropoulos, Nov 02 2021 *)
(* Second program generates a million terms in less than a minute: *)
c[_] = 0; c[1] = m = 1; {1}~Join~Reap[Do[Set[n, If[MissingQ[#], 2 m + 1, #]] &@ SelectFirst[Divisors[m], c[#] == 0 &]; c[n]++; Sow[n]; m = n, {i, 2^20}]][[-1, -1]]] (* Michael De Vlieger, Nov 05 2021 *)

Prog

(Python)
from sympy import divisors
terms = [1]
for i in range(100):
    for j in divisors(terms[-1]):
        if j not in terms:
            terms.append(j)
            break
    else:
        terms.append(terms[-1]*2+1)
print(terms) # Gleb Ivanov, Nov 09 2021

Crossrefs

Cf. A348871, A348873.

Sequence in context: A302029 A109732 A349183 * A114396 A348483 A333556

Adjacent sequences: A348869 A348870 A348871 * A348873 A348874 A348875

Keyword

base,nonn

Author

Eric Angelini and Carole Dubois, Nov 02 2021

Status

approved