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

Author

Charles L. Hohn, Mar 21 2025

Keywords

• nonn

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).

Examples

			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.
		

Crossrefs

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

Subsequence of A025487.

Programs

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 && dmx, mx=m; print1(x, ", ")))