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A364814
Numbers k whose largest divisor <= sqrt(k) is a power of 2, listing only the first such number with any given prime signature.
0
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
1,2
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.
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
David A. Corneth, Oct 21 2023
EXTENSIONS
Edited by Peter Munn, Oct 26 2023
STATUS
approved