A364814 - OEIS

%I #28 Oct 28 2023 09:33:43

%S 1,2,4,6,8,16,20,24,32,64,72,80,96,128,256,288,320,336,384,512,1024,

%T 1056,1152,1280,1344,1536,2048,4096,4224,4608,4800,5120,5376,6144,

%U 8192,16384,16896,17280,18432,18816,19200,20480,21504,24576,32768,65536,67584,69120,69888

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

%C 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.

%C 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.

%e 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.

%o (PARI)

%o upto(n) = {

%o 	my(res = List([1]), m = Map());

%o 	forstep(i = 2, n, 2,

%o 		if(isok(i),

%o 			s = sig(i);

%o 			sb = sigback(s);

%o 			if(!mapisdefined(m, sb),

%o 				listput(res, i);	

%o 				mapput(m, sb, i)

%o 			)

%o 		)

%o 	);

%o 	res

%o 	

%o }

%o sig(n) = {

%o 	vecsort(factor(n)[,2],,4)

%o }

%o sigback(v) = {

%o 	my(pr = primes(#v));

%o 	prod(i = 1, #v, pr[i]^v[i])

%o }

%o isok(n) = my(d = divisors(n)); hammingweight(d[(#d + 1)\2]) == 1

%Y Cf. A025487, A212171, A365406.

%K nonn

%O 1,2

%A _David A. Corneth_, Oct 21 2023

%E Edited by _Peter Munn_, Oct 26 2023