A364814 Numbers k whose largest divisor <= sqrt(k) is a power of 2, listing only the first such number with any given prime signature.

1, 2, 4, 6, 8, 16, 20, 24, 32, 64, 72, 80, 96, 128, 256, 288, 320, 336, 384, 512, 1024, 1056, 1152, 1280, 1344, 1536, 2048, 4096, 4224, 4608, 4800, 5120, 5376, 6144, 8192, 16384, 16896, 17280, 18432, 18816, 19200, 20480, 21504, 24576, 32768, 65536, 67584, 69120, 69888

Offset: 1

Author

David A. Corneth, Oct 21 2023

Keywords

• nonn

Comments

This sequence is a primitive sequence related to A365406 in the sense that it can be used to find the smallest term k in A365406 such that tau(k), omega(k) or bigomega(k) has some particular value.

Not every prime signature produces a term. For example no term has prime signature (3, 2, 1). Proof: any number with prime signature (3, 2, 1) has 24 divisors. Hence the 12th divisor must be a power of 2. But the largest power of 2 such number can have as a divisor is 8. 8 can never be the 12th divisor of a number. Therefore (3, 2, 1) can never be the prime signature of a term.

Examples

			k = 20 = 2^2 * 5 is in the sequence as it has prime signature (2, 1) and its largest divisor <= sqrt(k) is 4, a power of 2. It is the smallest such number since smaller numbers with prime signature (2, 1), namely 12 and 18, do not have the relevant divisor being a power of 2.
		

Crossrefs

Cf. A025487, A212171, A365406.

Programs

PARI

upto(n) = {
	my(res = List([1]), m = Map());
	forstep(i = 2, n, 2,
		if(isok(i),
			s = sig(i);
			sb = sigback(s);
			if(!mapisdefined(m, sb),
				listput(res, i);	
				mapput(m, sb, i)
			)
		)
	);
	res
}
sig(n) = {
	vecsort(factor(n)[,2],,4)
}
sigback(v) = {
	my(pr = primes(#v));
	prod(i = 1, #v, pr[i]^v[i])
}
isok(n) = my(d = divisors(n)); hammingweight(d[(#d + 1)\2]) == 1

Extensions

Edited by Peter Munn, Oct 26 2023