A390536 - OEIS

%I #74 Apr 10 2026 01:04:14

%S 15,25,27,51,55,57,63,85,95,99,111,115,117,119,121,123,125,185,187,

%T 201,205,207,209,213,215,219,221,231,235,237,243,245,247,249,253,255,

%U 289,323,335,355,361,365,387,391,393,395,407,411,415,417,423,425,429,437,445,447,451

%N 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)).

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

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

%H Szymon Łukaszyk, <a href="https://doi.org/10.48550/arXiv.2601.15364">On the quantum separability of qubit registers</a>, arXiv:2601.15364 [physics.gen-ph], 2026.

%H Szymon Łukaszyk, <a href="https://doi.org/10.3389/frqst.2026.1754112">On the quantum separability of qubit registers</a>, Frontiers, 2026.

%e 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)).

%e 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)).

%o (MATLAB)

%o function k_belongs=isok(k)

%o   k_belongs=false;

%o   if mod(k, 2) %odd k

%o     f = factor(k);  % Prime factors

%o     n = numel(f);

%o     products = [];  % Generate all possible subsets (except empty) Using binary masks

%o     for i = 1:2^n - 1

%o       subset = f(logical(bitget(i, 1:n)));

%o       b = prod(subset);

%o       c = k / b;

%o       if mod(k, b) == 0 & b>1 & c>1

%o         products = [products; sort([b c])];

%o       end

%o     end

%o     products = unique(products, 'rows');  % Remove duplicate products

%o     if length(products)  % k is not a prime

%o       seq_candidate = 0;

%o       for l=1:size(products,1)

%o         if ceil(log2(products(l,1))) + ceil(log2(products(l,2))) > ceil(log2(k)) % k=l*m is a candidate

%o           seq_candidate = seq_candidate+1;

%o         end

%o       end

%o       if seq_candidate == size(products,1) % All possible products of k=b*c are candidates

%o         k_belongs=true;

%o       end

%o     end

%o   end

%o end

%o (PARI) c(n) = if( n<2, 0, exponent(n-1)+1); \\ A029837

%o 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

%o (Python)

%o from itertools import count, islice, takewhile

%o from sympy import isprime, divisors

%o def A390536_gen(startvalue=4): # generator of terms >= startvalue

%o     for n in count(max(4,startvalue)):

%o         m = (n-1).bit_length()

%o         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))):

%o             yield n

%o A390536_list = list(islice(A390536_gen(),57)) # _Chai Wah Wu_, Nov 20 2025

%Y Cf. A003313, A014701, A029837, A131288.

%K nonn

%O 1,1

%A _Szymon Lukaszyk_, Nov 09 2025

%E More terms from _Michel Marcus_, Nov 11 2025