A387153
Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.
Original entry on oeis.org
30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 79170, 82110, 84630, 85470, 91770, 94710, 99330, 101010, 103530, 108570, 111930, 117390, 122430, 128310, 136290, 140910, 144690, 154770, 161070, 164010, 166530, 168630, 182490, 191730, 205590
Offset: 1
30030 = 2 * 3 * 5 * 7 * 11 * 13 is a term since it is squarefree, and sigma(30030) = 96768 > 3*30030 = 90090.
-
q[k_] := Module[{f = FactorInteger[k]}, Max[f[[;;, 2]]] == 1 && Times @@ (1 + f[[;; , 1]]) > 3*k]; Select[Range[2*10^5], q]
-
isok(k) = {my(f = factor(k)); issquarefree(f) && vecprod(apply(x -> x+1, f[, 1])) > 3*k;}
A359127
Oblong numbers which are products of six distinct primes.
Original entry on oeis.org
43890, 53130, 81510, 108570, 152490, 184470, 188790, 260610, 297570, 371490, 416670, 475410, 509082, 549822, 593670, 637602, 648830, 756030, 757770, 814506, 932190, 939930, 973182, 1003002, 1045506, 1135290, 1178310, 1222130, 1233210, 1257762, 1278030, 1332870, 1414910, 1417290, 1484742
Offset: 1
43890 = 209*210 = 2*3*5*7*11*19
53130 = 230*231 = 2*3*5*7*11*23
81510 = 285*286 = 2*3*5*11*13*19
108570 = 329*330 = 2*3*5*7*11*47
-
R:= NULL: count:= 0:
for n from 1 while count < 100 do
x:= n*(n+1);
F:= ifactors(x)[2];
if nops(F) = 6 and max(map(t -> t[2],F))=1 then
R:= R, x; count:= count+1;
fi
od:
R; # Robert Israel, Dec 26 2022
-
Select[(#*(# + 1)) & /@ Range[1250], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1, 1} &] (* Amiram Eldar, Dec 26 2022 *)
A359645
Number of numbers <= 10^n that are products of 6 distinct primes.
Original entry on oeis.org
0, 0, 0, 0, 20, 1235, 32396, 605939, 9446284, 131733664, 1706815354, 21008871506, 249145286508, 2873325692759, 32433194803107, 359960491516138, 3941261642520039, 42679704453671033, 457980431402674541
Offset: 1
a(5) = 20 = #{30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 72930, 79170, 81510, 82110, 84630, 85470, 91770, 94710, 98670, 99330}.
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a(n) = my(N=10^n); (f(m, p, k, j=1)=my(s=sqrtnint(N\m, k), count=0); if(k==2, forprime(q=p, s, count += primepi(N\(m*q)) - j; j+=1); return(count)); forprime(q=p, s, count += f(m*q, q+1, k-1, j+1); j+=1); count); f(1, 2, 6); \\ Daniel Suteu, Jan 11 2023
A362758
Triangular numbers which are products of six distinct primes.
Original entry on oeis.org
207690, 255255, 274170, 303810, 304590, 323610, 370230, 391170, 426426, 487578, 649230, 650370, 744810, 763230, 856086, 951510, 1007490, 1186570, 1248990, 1352190, 1365378, 1376970, 1473186, 1512930, 1528626, 1567335, 1594005, 1655290, 1657110, 1747515, 1775670, 1911990, 1991010, 2003001
Offset: 1
a(1) = T(644) = 644*(644+1)/2 = 207690 = 2*3*5*7*23*43.
a(2) = T(714) = 714*(714+1)/2 = 255255 = 3*5*7*11*13*17.
a(3) = T(740) = 740*(740+1)/2 = 274170 = 2*3*5*13*19*37.
-
Select[Accumulate[Range[2000]], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1, 1} &] (* Amiram Eldar, May 02 2023 *)
A379167
Table read by row, where T(n,k), n>0 and k>0, represents the smallest n-digit number that is the product of k distinct primes and is sandwiched between semiprime numbers, or -1 if no such number exists.
Original entry on oeis.org
5, -1, 34, -1, 122, 186, 870, -1, 1042, 1146, 1190, 5610, -1, 10118, 10002, 10030, 10230, 39270, -1, 100462, 100158, 100030, 100122, 110670, 881790, -1, 1000478, 1000022, 1000010, 1000758, 1001130, 1009470, -1, 10000202, 10000258, 10000002, 10000218, 10001670, 10010910, 15825810
Offset: 1
Smallest n-digit numbers m product of k distinct primes sandwiched between two semiprimes m-1 and m+1:
n k m-1, m, m+1
1 1 4 = 2^2, 5 = 5, 6 = 2 * 3;
2 2 33 = 3 * 11, 34 = 2 * 17, 35 = 5 * 7;
3 3 185 = 5 * 37, 186 = 2 * 3 * 31, 187 = 11 * 17;
Table read by rows:
5;
-1, 34;
-1, 122, 186, 870;
-1, 1042, 1146, 1190, 5610;
-1, 10118, 10002, 10030, 10230, 39270;
-1, 100462, 100158, 100030, 100122, 110670, 881790;
-1, 1000478, 1000022, 1000010, 1000758, 1001130, 1009470;
-1, 10000202, 10000258, 10000002, 10000218, 10001670, 10010910, 15825810;
...
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