A354255 Even numbers in A090252 in order of appearance.

2, 4, 8, 16, 26, 32, 64, 128, 206, 256, 478, 512, 998, 1024, 2048, 3134, 4096, 6514, 8192, 13942, 16384, 28894, 32768, 60518, 65536, 126634, 131072, 261398, 262144

Offset: 1

Author

Michael S. Branicky, May 21 2022

Keywords

• nonn
• more

Comments

The n-th even term in A090252 appears at index k <= A083329(n).

Conjecture: The indices of even numbers in A090252 are precisely the numbers {A083329(n), n >= 1}. See A090252 for discussion. - N. J. A. Sloane, May 22 2022

Taking logs to base 2 of these terms produces 1., 2., 3., 4., 4.700439718, 5., 6., 7., 7.686500527, 8., 8.900866807, 9., 9.962896004, 10., 11., 11.61378946, 12., 12.66932800, 13., 13.76714991, 14. - N. J. A. Sloane, Jun 01 2022

Crossrefs

Cf. A083329, A090252, A247665, A248379, A353730, A354146, A354159.

Programs

Python

from math import gcd, prod
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, mink = [1], {1}, 2
    for n in count(2):
        k, s = mink, n - n//2
        prodall = prod(alst[n-n//2-1:n-1])
        while k in aset or gcd(prodall, k) != 1: k += 1
        alst.append(k); aset.add(k)
        if k%2 == 0: yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 9))) # Michael S. Branicky, May 23 2022

Extensions

a(14) from Michael S. Branicky, May 26 2022

a(15)-a(21) from Michael S. Branicky, Jun 01 2022 using gzipped b-file in A090252

a(22)-a(26) from Hugo van der Sanden, Jun 14 2022

a(27)-a(29) from Jinyuan Wang, Jul 15 2022