cp's OEIS Frontend

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

%I A386972 #26 Aug 17 2025 21:51:19
%S A386972 96,144,160,224,240,324,336,352,360,400,416,486,504,528,540,544,560,
%T A386972 600,608,624,736,756,784,792,810,816,880,900,912,928,936,992,1040,
%U A386972 1104,1134,1176,1184,1188,1215,1224,1232,1260,1312,1350,1360,1368,1376,1392,1400,1404,1456,1488,1500
%N A386972 Numbers that are the product of a semiprime and the square of another semiprime.
%C A386972 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
%H A386972 David A. Corneth, <a href="/A386972/a386972.png">First 1000000 terms colorcoded by prime signature</a>
%e A386972 96 = 6 * 4^2;
%e A386972 144 = 9 * 4^2 or 4 * 6^2.
%p A386972 M:= 2000: # for terms <= M
%p A386972 P:= select(isprime, [2,seq(i,i=3..M/8,2)]): nP:= nops(P):
%p A386972 S:= {}:
%p A386972 for i1 from 1 to nP do
%p A386972   p1:= P[i1];
%p A386972   if p1^2*4^2 > M then break fi;
%p A386972   for i2 from i1 to nP do
%p A386972     p2:= P[i2];
%p A386972     if p1*p2*4^2 > M then break fi;
%p A386972     for i3 from 1 to nP do
%p A386972       p3:= P[i3];
%p A386972       if p1*p2*p3^4 > M then break fi;
%p A386972       for i4 from i3 to nP do
%p A386972         p4:= P[i4];
%p A386972         v:= p1*p2*(p3*p4)^2;
%p A386972         if v > M then break fi;
%p A386972         if p1*p2 = p3*p4 then next fi;
%p A386972         S:= S union {v}
%p A386972 od od od od:
%p A386972 sort(convert(S,list)); # _Robert Israel_, Aug 11 2025
%t A386972 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 *)
%o A386972 (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
%Y A386972 Cf. A001358 (semiprimes), A046306, A054753, A386977.
%K A386972 nonn
%O A386972 1,1
%A A386972 _Ian Hahus_, Aug 11 2025