A379762 Products of 4 distinct prime numbers (or tetraprimes) that are abundant.

210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, 1190, 1218, 1230, 1254, 1290, 1302, 1326, 1330, 1410, 1430, 1482, 1518, 1554, 1590, 1610, 1722, 1770, 1794, 1806, 1830, 1870, 1914, 1938, 1974, 2002, 2010, 2030, 2046, 2090, 2130, 2170, 2190, 2210, 2226, 2262, 2346, 2370, 2418, 2442, 2470, 2478, 2490, 2530, 2562, 2590, 2622

Offset: 1

Author

Massimo Kofler, Jan 09 2025

Keywords

• nonn

Comments

This sequence is not 2*{A046389}. 2618 = 2*1309 is not in this sequence, while 1309 is in A046389.

Contains 6*p*q if p and q are distinct primes > 3. The first term not of this form is 770. - Robert Israel, Jan 09 2025

a(43) = 2002 is the only term coprime to 15. - Charles R Greathouse IV, Jan 13 2025

Examples

			210 is a term because 210=2*3*5*7 is the product of four distinct primes and it is smaller than the sum of its proper divisors 366.
1155 is not a term because 1155=3*5*7*11 is the product of four distinct primes and it is larger than the sum of its proper divisors 1149.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005101 and A046386.

Cf. A001065, A378717, A379917.

Programs

Maple

filter:= proc(n) local F,t;
   F:= ifactors(n)[2];
   F[..,2] = [1,1,1,1] and mul(t[1]+1, t = F) > 2*n
end proc:
select(filter, [seq(i,i=2..3000, 4)]); # Robert Israel, Jan 09 2025

Mathematica

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1, 1} && Times @@ (1 + 1/f[[;; , 1]]) > 2]; Select[Range[3000], q] (* Amiram Eldar, Jan 09 2025 *)

PARI

list(lim)=my(v=List(select(k->k<=lim, [1430, 1870, 2002, 2090, 2210, 2470, 2530, 2990, 3190, 3230, 3410, 3770, 4030, 4070, 4510, 4730, 5170, 5830]))); forprime(p=5,sqrtint(lim\6), my(t=6*p); forprime(q=p+2,lim\t, listput(v,t*q))); forprime(p=11,lim\70,listput(v,70*p)); Set(v) \\ Charles R Greathouse IV, Jan 13 2025

Formula

a(n) == 2 (mod 4).

a(n) ~ (1/6)*n log n/log log n. - Charles R Greathouse IV, Jan 13 2025