A382953 Numbers with at least one factorization for which the factors can be partitioned into 2 or more distinct subsets with equal sums.

16, 30, 48, 54, 64, 70, 72, 84, 96, 120, 126, 128, 144, 160, 162, 180, 192, 198, 210, 216, 240, 243, 250, 252, 256, 264, 270, 280, 286, 288, 300, 308, 320, 324, 330, 336, 360, 378, 384, 390, 396, 400, 420, 432, 440, 448, 462, 468, 480, 486, 495, 504, 510, 512

Offset: 1

Author

Charles L. Hohn, Apr 09 2025

Keywords

• nonn

Comments

Here "distinct" means that no partition contains the same subset of factors, e.g. 4 is not a term because {2} == {2}.

Because 2 + 2 = 2 * 2 = 4, many terms have multiple instances that differ only by factors {2, 2} vs. {4}, except in some cases where such substitutions would create indistinct subsets, e.g. while 16 is a term for partition set {{2, 2}, {4}}, {{2, 2}, {2, 2}} and {{4}, {4}} do not count as additional instances.

For primes p and integers x >= 0, p^(p+2+2x) and p^(2p+3+x) are terms.

For integers x and y >= 0, (4x+4)^(y+2) and (4x+6)^(y+3) are terms.

First few terms with record counts of instances: 16 (1 instance), 48 (2), 120 (3), 240 (6), 576 (8), 720 (9), 768 (12).

If k is a term, then 4k is also a term. - Ivan N. Ianakiev, Apr 10 2025

Examples

			a(1) = 16: 2 * 2 * 4 = 16 and 2 + 2 = 4.
a(2) = 30: 2 * 3 * 5 = 30 and 2 + 3 = 5.
a(3) = 48: 2 * 2 * 2 * 6 = 48 and 2 + 2 + 2 = 6, and also 2 * 4 * 6 = 48 and 2 + 4 = 6.
a(5) = 64: 2 * 2 * 2 * 2 * 4 = 64 and 2 + 2 + 2 = 2 + 4.
a(39) = 384: 2 * 2 * 2 * 2 * 4 * 6 = 384 and 2 + 2 + 2 = 2 + 4 = 6 (plus 4 other instances).
		

Crossrefs

Cf. A083207, A322657, A255265 (subsequence).

Programs

Mathematica

ok[n_]:=Catch@ Block[{t, d=Divisors@n,f}, f[y_]:=Block[{L={}, r}, r[x_,m_,c_]:= If[x==1, AppendTo[L,c], r[x/#, #, Append[c,#]]& /@ Select[ Divisors@x, #>=m&];]; f[y,2,{}]; L]; Do[t=Plus@@@ s[d[[i]]]; If[d[[i]]^2!=n, Intersection[t, Plus@@@ s[n/d[[i]]]] != {} && Throw@True, Sort@t != Union@t && Throw@True],{i, 2, Ceiling[ Length@d/2]}]; False]; Select[Range@ 512,ok] (* Giovanni Resta, Apr 10 2025 *)

PARI

a382953_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(#f && r==1, my(c=0, s=vecsum(Vec(f)), d=divisors(s)); for(i=2, #d, my(z=s/d[i]); if(z1, next); listput(f, d); c+=a382953_count(x, f); listpop(f)); return(c)}
a382953_part(f, z, rvs=0, v=List())={my(c=0); if(#v==#f[2], if(sum(i=1, #v, f[1][i]*v[i])