A365476 a(n) is the minimum of A000120(k)*A000120(A071904(n)/k) for divisors k of the n-th odd composite number A071904(n) other than 1 and A071904(n).

4, 4, 6, 4, 4, 6, 6, 6, 4, 9, 4, 6, 6, 6, 6, 8, 6, 9, 4, 4, 8, 9, 10, 6, 4, 6, 6, 8, 6, 6, 9, 6, 6, 8, 9, 8, 10, 9, 8, 6, 4, 10, 8, 12, 4, 9, 6, 6, 10, 10, 6, 6, 6, 4, 6, 12, 6, 6, 9, 8, 8, 15, 6, 6, 6, 6, 10, 10, 6, 6, 9, 8, 12, 8, 9, 8, 8, 8, 9, 9, 10, 8, 9, 4, 6, 10, 4, 12, 12, 8, 10, 10, 6

Offset: 1

Author

Robert Israel, Sep 04 2023

Keywords

• nonn
• base

Comments

a(n) = 4 iff A071904(n) is the product of two (not necessarily distinct) members of A000051.

a(n) >= A000120(A071904(n)) because A000120(x) * A000120(y) >= A000120(x*y).

a(n) <= A349494(A071904(n)).

Examples

			a(9) = 4 because A071904(9) = 45 = 3 * 15 = 5 * 9 with A000120(3) * A000120(15) = 2 * 4 = 8 and A000120(5) * A000120(9) = 2 * 2 = 4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000051, A000120, A071904, A349494.

Programs

Maple

g:= proc(n) convert(convert(n, base, 2), `+`) end proc:
f:= proc(n) local t, r;
      min(seq(g(t)*g(n/t), t = numtheory:-divisors(n) minus {1,n}))
    end proc:
map(f, remove(isprime, [seq(i,i=3..1000,2)]));

Python

from sympy import primepi, divisors
def A365476(n):
    if n == 1: return 4
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return min(int(d).bit_count()*int(m//d).bit_count() for d in divisors(m,generator=True) if 1Chai Wah Wu, Aug 02 2024