A371699 The smallest composite number which divides the concatenation of its descending ordered prime factors, with repetition, when written in base n.

42, 4, 42, 4, 374, 4, 9, 4, 378, 4, 609, 4, 9, 4, 3525, 4, 343, 4, 9, 4, 70, 4, 25, 4, 9, 4, 195, 4, 343, 4, 9, 4, 25, 4, 130, 4, 9, 4, 366, 4, 3562, 4, 9, 4, 42, 4, 49, 4, 9, 4, 474, 4, 25, 4, 9, 4, 238, 4, 1131, 4, 9, 4, 25, 4, 555, 4, 9, 4, 14405, 4, 12207

Offset

2,1

Comments

Conjecture: a(n) <= A371641(n) for n >= 2.

Links

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Formula

If p is prime, then a(p*(m+2)-1) <= p^2 for all m >= 0.

If n+1 is composite, then a(n) <= q^2, where q = A020639(n+1) is the smallest prime factor of n+1. This implies that if n > 2 is odd, then a(n) = A371641(n) = 4.

The first few terms n where n+1 is composite and a(n) < A020639(n+1)^2 are a(288) = 70, a(298) = 42, a(340) = 42, a(360) = 182, ...

If n is even, then a(n) >= 9. This is true as it is easy to verify that a(n) cannot be equal to 4, 6 or 8 in this case.

Suppose n>2 is even. 2 concatenated twice in base n is 2(n+1) which is not divisible by 4.

Next, 3 concatenated by 2 is 3*n+2 which is not divisible by 6. Finally 2 concatenated 3 times is 2(n^3-1)/(n-1) which is not divisible by 8 since n^3-1 is odd.

This implies that if n = 6*k+2 for some k > 0, then a(n) = A371641(n) = 9.

Example

a(2) = 42 since 42 = 7*3*2 = 111_2 * 11_2 * 10 _2 and 42 divides 126 = 1111110_2.
a(10) = 378 since 278 = 7*3*3*3*2 and 278 divides 73332.

Prog

(Python)
from itertools import count
from sympy import factorint, integer_log
def A371699(n):
    for m in count(4):
        f = factorint(m)
        if sum(f.values()) > 1:
            c = 0
            for p in sorted(f, reverse=True):
                a = pow(n, integer_log(p, n)[0]+1, m)
                for _ in range(f[p]):
                    c = (c*a+p)%m
            if not c:
                return m

Crossrefs

Cf. A020639, A027746, A259047, A322843, A248915, A371641, A371821.

Sequence in context: A203484 A176920 A037938 * A091747 A030434 A194710

Adjacent sequences: A371696 A371697 A371698 * A371700 A371701 A371702

Keyword

nonn,base

Author

Chai Wah Wu, Apr 12 2024

Status

approved