A378717 Products of 4 distinct prime numbers (or tetraprimes) that are deficient.

1155, 1365, 1785, 1995, 2145, 2415, 2618, 2805, 2926, 3003, 3045, 3094, 3135, 3255, 3315, 3458, 3542, 3705, 3795, 3885, 3910, 3927, 4186, 4305, 4370, 4389, 4466, 4485, 4515, 4522, 4641, 4774, 4785, 4810, 4845, 4862, 4930, 4935, 5005, 5115, 5187, 5270, 5278, 5313, 5330, 5434, 5474, 5510, 5565, 5590

Offset: 1

Author

Massimo Kofler, Dec 05 2024

Keywords

• nonn

Examples

			1155 is 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 (1+3+5+7+11+15+21+33+35+55+77+105+165+231+385=1149).
1365 is a term because 1365=3*5*7*13 is the product of four distinct primes and it is larger than the sum of its proper divisors (1+3+5+7+13+15+21+35+39+65+91+105+195+273+455=1323).
		

Crossrefs

Intersection of A005100 and A046386. A046390 is a subsequence.

Cf. A378480.

Programs

Mathematica

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1, 1} && Times @@ (1 + 1/f[[;; , 1]]) < 2]; Select[Range[6000], q] (* Amiram Eldar, Dec 05 2024 *)

PARI

catpr(~v, lim, mult, startAt)=forprime(p=startAt,lim\mult, listput(v,mult*p))
list(lim)=my(v=List()); forprime(p=3,sqrtnint(lim\=1,4), forprime(q=p+2,sqrtnint(lim\p,3), forprime(r=q+2,sqrtint(lim\p\q), catpr(~v,lim,p*q*r, r+2)))); forprime(p=11,sqrtnint(lim\2,3), forprime(q=13,sqrtint(lim\2\p), catpr(~v, lim, 2*p*q, q+2))); forprime(p=13,sqrtint(lim\14), catpr(~v,lim,14*p,p+2)); forprime(p=19,sqrtint(lim\10), catpr(~v,lim, 10*p, p+2)); catpr(~v, lim, 154, 17); catpr(~v, lim, 110, 59); catpr(~v, lim, 130, 37); catpr(~v, lim, 170, 23); Set(v) \\ Charles R Greathouse IV, Dec 06 2024

Formula

a(n) ~ A046390(n) ~ A046386(n) ~ A014613(n) ~ 6n log n / (log log n)^3. - Charles R Greathouse IV, Dec 06 2024