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