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A345395 Composite numbers whose divisors that are larger than 1 are all digitally balanced numbers in base 2 (A031443).

%I A345395 #15 Jul 03 2025 01:02:21
%S A345395 132061,138421,151427,532393,545269,546407,557983,559609,568801,
%T A345395 570709,573193,579013,590687,595853,599707,604873,610777,624553,
%U A345395 630293,635213,2102767,2105063,2109383,2111339,2123677,2128187,2129081,2129609,2143961,2149753,2151131,2151661
%N A345395 Composite numbers whose divisors that are larger than 1 are all digitally balanced numbers in base 2 (A031443).
%C A345395 The prime numbers with this property are the digitally balanced primes (A066196).
%C A345395 All the terms are odd, since if k is an even digitally balanced number then its divisor k/2 is not digitally balanced (since it has one fewer 0 in its binary expansion).
%C A345395 Apparently most of the terms are semiprimes (A001358) with 4 divisors.
%C A345395 Terms with 3 divisors, i.e., squares of primes: 145178401 = 12049^2, 155575729 = 12473^2, ...
%C A345395 The least term with more than 4 divisors is 8897396239 = 163 * 929 * 58757, with 8 divisors.
%C A345395 The least term with 6 divisors is 8923691369 = 41 * 14753^2.
%H A345395 Amiram Eldar, <a href="/A345395/b345395.txt">Table of n, a(n) for n = 1..10000</a>
%e A345395 132061 is a term since its divisors that are larger than 1 are {41, 3221, 132061}, and their binary representations are {101001, 110010010101, 100000001111011101}. Each one has an equal number of 0's and 1's.
%t A345395 balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; Select[Range[9,10^6,2], CompositeQ[#] && AllTrue[Rest@Divisors[#], balQ] &]
%o A345395 (PARI) isbal(k) = exponent(k) + 1 == 2 * hammingweight(k);
%o A345395 isok(k) = if(k == 1 || isprime(k), 0, fordiv(k, d, if(d > 1 && !isbal(d), return(0))); 1); \\ _Amiram Eldar_, Jul 03 2025
%Y A345395 Subsequence of A031443.
%Y A345395 Cf. A066196.
%K A345395 nonn,base
%O A345395 1,1
%A A345395 _Amiram Eldar_, Jun 17 2021