A382325 Numbers with a record ratio of proper factorizations to nontrivial divisors.

4, 16, 32, 64, 128, 192, 256, 384, 512, 576, 768, 864, 1024, 1152, 1536, 1728, 2304, 3456, 4608, 5184, 5760, 6912, 8640, 9216, 10368, 11520, 13824, 17280, 20736, 23040, 25920, 27648, 34560, 41472, 51840, 62208, 69120, 82944, 103680, 138240, 165888, 172800

Offset

1,1

Comments

Numbers k that give a record value for A028422(k)/A070824(k).

a(n) = 0 (mod 4), and with prime factors of terms clustering around the smallest primes, it is observed that as n increases, the gcd of a(n)..a(oo) remains among the largest divisors of a(n).

Links

Table of n, a(n) for n=1..42.

Example

a(1)=4: |{{2, 2}}| / |{2}| = 1/1.
a(2)=16: |{{2, 2, 2, 2}, {2, 2, 4}, {2, 8}, {4, 4}}| / |{2, 4, 8}| = 4/3.
a(3)=32: |{{2, 2, 2, 2, 2}, {2, 2, 2, 4}, {2, 2, 8}, {2, 4, 4}, {2, 16}, {4, 8}}| / |{2, 4, 8, 16}| = 6/4.

Prog

(PARI) f_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(#f, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && d<f[#f]), next); listput(f, d); c+=f_count(x, f); listpop(f)); return(c)}
my(mx=0); for(x=1, 200000, my(d=numdiv(x)-2); if(!d, next); my(m=f_count(x)/d); if(m>mx, mx=m; print1(x, ", ")))

Crossrefs

Cf. A028422, A070824, A033833, A002182, A382326, A382327.

Subsequence of A025487.

Sequence in context: A326873 A126032 A391762 * A296819 A034713 A374001

Adjacent sequences: A382322 A382323 A382324 * A382326 A382327 A382328

Keyword

nonn

Author

Charles L. Hohn, Mar 21 2025

Status

approved