A371821 Composite numbers which divide the concatenation of their ascending ordered prime factors, with repetition, when written in binary.

85329, 177904587, 333577497

Offset: 1

Author

Scott R. Shannon, Apr 07 2024

Keywords

• nonn
• base
• more
• bref

Comments

The base 2 version of A259047. Assuming a(4) exists it is greater than 10^10.

a(4) <= 55133857902732922904331439521901. - Chai Wah Wu, Apr 12 2024

a(1), a(3), the bound on a(4) above, and larger terms can be generated using an adaptation of the method of J. K. Andersen referenced in A259047; see linked Python program for an implementation and two more terms. - Michael S. Branicky, Apr 12 2024

Examples

			177904587 is a term as 177904587 = 3_10 * 7_10 * 103_10 * 233_10 * 353_10 = 11_2 * 111_2 * 1100111_2 * 11101001_2 * 101100001_2 = "11111110011111101001101100001"_2 = 533713761_10, which is divisible by 177904587.
		

Links

• Michael S. Branicky, Python program generating terms in A371821

Crossrefs

Cf. A027746, A004676, A259047, A371641.

Programs

Python

from itertools import count, islice
from sympy import factorint
def A371821_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = sorted(factorint(n,multiple=True))
        if len(f) > 1:
            c = 0
            for p in f:
                c = ((c<A371821_list = list(islice(A371821_gen(),3)) # Chai Wah Wu, Apr 11 2024

Python

from sympy import factorint, isprime
def ok(n): return not isprime(n) and int("".join(bin(p)[2:]*e for p, e in factorint(n).items()), 2)%n == 0 # Michael S. Branicky, Apr 12 2024