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

132061, 138421, 151427, 532393, 545269, 546407, 557983, 559609, 568801, 570709, 573193, 579013, 590687, 595853, 599707, 604873, 610777, 624553, 630293, 635213, 2102767, 2105063, 2109383, 2111339, 2123677, 2128187, 2129081, 2129609, 2143961, 2149753, 2151131, 2151661

Offset: 1

Author

Amiram Eldar, Jun 17 2021

Keywords

• nonn
• base

Comments

The prime numbers with this property are the digitally balanced primes (A066196).

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).

Apparently most of the terms are semiprimes (A001358) with 4 divisors.

Terms with 3 divisors, i.e., squares of primes: 145178401 = 12049^2, 155575729 = 12473^2, ...

The least term with more than 4 divisors is 8897396239 = 163 * 929 * 58757, with 8 divisors.

The least term with 6 divisors is 8923691369 = 41 * 14753^2.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A031443.

Cf. A066196.

Programs

Mathematica

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] &]

PARI

isbal(k) = exponent(k) + 1 == 2 * hammingweight(k);
isok(k) = if(k == 1 || isprime(k), 0, fordiv(k, d, if(d > 1 && !isbal(d), return(0))); 1); \\ Amiram Eldar, Jul 03 2025