A386972 Numbers that are the product of a semiprime and the square of another semiprime.

96, 144, 160, 224, 240, 324, 336, 352, 360, 400, 416, 486, 504, 528, 540, 544, 560, 600, 608, 624, 736, 756, 784, 792, 810, 816, 880, 900, 912, 928, 936, 992, 1040, 1104, 1134, 1176, 1184, 1188, 1215, 1224, 1232, 1260, 1312, 1350, 1360, 1368, 1376, 1392, 1400, 1404, 1456, 1488, 1500

Offset: 1

Author

Ian Hahus, Aug 11 2025

Keywords

• nonn

Comments

Numbers with prime signature [5, 1], [4, 2], [4, 1, 1], [3, 2, 1], [2, 2, 2] or [2, 2, 1, 1]. So, necessarily but not sufficiently, terms t have bigomega(t) = 6. - David A. Corneth, Aug 11 2025

Examples

			96 = 6 * 4^2;
144 = 9 * 4^2 or 4 * 6^2.
		

Links

• David A. Corneth, First 1000000 terms colorcoded by prime signature

Crossrefs

Cf. A001358 (semiprimes), A046306, A054753, A386977.

Programs

Maple

M:= 2000: # for terms <= M
P:= select(isprime, [2,seq(i,i=3..M/8,2)]): nP:= nops(P):
S:= {}:
for i1 from 1 to nP do
  p1:= P[i1];
  if p1^2*4^2 > M then break fi;
  for i2 from i1 to nP do
    p2:= P[i2];
    if p1*p2*4^2 > M then break fi;
    for i3 from 1 to nP do
      p3:= P[i3];
      if p1*p2*p3^4 > M then break fi;
      for i4 from i3 to nP do
        p4:= P[i4];
        v:= p1*p2*(p3*p4)^2;
        if v > M then break fi;
        if p1*p2 = p3*p4 then next fi;
        S:= S union {v}
od od od od:
sort(convert(S,list)); # Robert Israel, Aug 11 2025

Mathematica

Select[Range@ 1500, MemberQ[{{1,5}, {2,4}, {1,1,4}, {1,2,3}, {2,2,2}, {1,1,2,2}}, Sort[ Last /@ FactorInteger[#]]] &] (* Giovanni Resta, Aug 12 2025 *)

PARI

is(n) = {my(f = factor(n), b = bigomega(f)); if(b != 6, return(0)); f = vecsort(f[,2]~); #setminus(Set([f]), Set([[1, 5], [2, 4], [1, 1, 4], [1, 2, 3], [2, 2, 2], [1, 1, 2, 2]])) == 0} \\ David A. Corneth, Aug 12 2025