cp's OEIS Frontend

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

%I A364814 #31 Dec 27 2024 00:57:25
%S A364814 1,2,4,6,8,16,20,24,32,64,72,80,96,128,256,288,320,336,384,512,1024,
%T A364814 1056,1152,1280,1344,1536,2048,4096,4224,4608,4800,5120,5376,6144,
%U A364814 8192,16384,16896,17280,18432,18816,19200,20480,21504,24576,32768,65536,67584,69120,69888
%N A364814 Numbers k whose largest divisor <= sqrt(k) is a power of 2, listing only the first such number with any given prime signature.
%C A364814 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 A364814 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 A364814 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 A364814 (PARI)
%o A364814 upto(n) = {
%o A364814 	my(res = List([1]), m = Map());
%o A364814 	forstep(i = 2, n, 2,
%o A364814 		if(isok(i),
%o A364814 			s = sig(i);
%o A364814 			sb = sigback(s);
%o A364814 			if(!mapisdefined(m, sb),
%o A364814 				listput(res, i);	
%o A364814 				mapput(m, sb, i)
%o A364814 			)
%o A364814 		)
%o A364814 	);
%o A364814 	res
%o A364814 }
%o A364814 sig(n) = {
%o A364814 	vecsort(factor(n)[,2],,4)
%o A364814 }
%o A364814 sigback(v) = {
%o A364814 	my(pr = primes(#v));
%o A364814 	prod(i = 1, #v, pr[i]^v[i])
%o A364814 }
%o A364814 isok(n) = my(d = divisors(n)); hammingweight(d[(#d + 1)\2]) == 1
%Y A364814 Cf. A025487, A212171, A365406.
%K A364814 nonn
%O A364814 1,2
%A A364814 _David A. Corneth_, Oct 21 2023
%E A364814 Edited by _Peter Munn_, Oct 26 2023