A390536 Composite integers k = b * c such that A029837(k) < A029837(b) + A029837(c) for all b>1, c>1, where A029837(x) = ceiling(log_2(x)).

15, 25, 27, 51, 55, 57, 63, 85, 95, 99, 111, 115, 117, 119, 121, 123, 125, 185, 187, 201, 205, 207, 209, 213, 215, 219, 221, 231, 235, 237, 243, 245, 247, 249, 253, 255, 289, 323, 335, 355, 361, 365, 387, 391, 393, 395, 407, 411, 415, 417, 423, 425, 429, 437, 445, 447, 451

Offset

1,1

Comments

These numbers cannot represent the support sizes of bipartite separable quantum registers containing ceiling(log_2(k)) qubits.

a(1)=15 is also the smallest n for which A003313(15) < A014701(15).

Links

Table of n, a(n) for n=1..57.

Szymon Łukaszyk, On the quantum separability of qubit registers, arXiv:2601.15364 [physics.gen-ph], 2026.

Szymon Łukaszyk, On the quantum separability of qubit registers, Frontiers, 2026.

Example

117 is a term since its two factorizations 117 = 3*39 and 9*13 both have ceiling(log_2(117)) = 7 < ceiling(log_2(b)) + ceiling(log_2(c)).
45 is not a term since among its factorizations is 45 = 3*15 which fails since ceiling(log_2(45)) >= ceiling(log_2(3)) + ceiling(log_2(15)).

Prog

(MATLAB)
function k_belongs=isok(k)
  k_belongs=false;
  if mod(k, 2) %odd k
    f = factor(k);  % Prime factors
    n = numel(f);
    products = [];  % Generate all possible subsets (except empty) Using binary masks
    for i = 1:2^n - 1
      subset = f(logical(bitget(i, 1:n)));
      b = prod(subset);
      c = k / b;
      if mod(k, b) == 0 & b>1 & c>1
        products = [products; sort([b c])];
      end
    end
    products = unique(products, 'rows');  % Remove duplicate products
    if length(products)  % k is not a prime
      seq_candidate = 0;
      for l=1:size(products, 1)
        if ceil(log2(products(l, 1))) + ceil(log2(products(l, 2))) > ceil(log2(k)) % k=l*m is a candidate
          seq_candidate = seq_candidate+1;
        end
      end
      if seq_candidate == size(products, 1) % All possible products of k=b*c are candidates
        k_belongs=true;
      end
    end
  end
end
(PARI) c(n) = if( n<2, 0, exponent(n-1)+1); \\ A029837
isok(k) = if (numdiv(k) > 2, fordiv(k, d, if ((d>1) && (d<k) && (c(k) >= c(d) + c(k/d)), return(0))); return(1)); \\ Michel Marcus, Nov 14 2025
(Python)
from itertools import count, islice, takewhile
from sympy import isprime, divisors
def A390536_gen(startvalue=4): # generator of terms >= startvalue
    for n in count(max(4, startvalue)):
        m = (n-1).bit_length()
        if not isprime(n) and all(d==1 or m<(d-1).bit_length()+(n//d-1).bit_length() for d in takewhile(lambda d:d*d<=n, divisors(n))):
            yield n
A390536_list = list(islice(A390536_gen(), 57)) # Chai Wah Wu, Nov 20 2025

Crossrefs

Cf. A003313, A014701, A029837, A131288.

Sequence in context: A325571 A383363 A366926 * A386307 A102802 A050692

Adjacent sequences: A390533 A390534 A390535 * A390537 A390538 A390539

Keyword

nonn

Author

Szymon Lukaszyk, Nov 09 2025

Extensions

More terms from Michel Marcus, Nov 11 2025

Status

approved