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%I #11 Dec 06 2024 16:31:40
%S 1155,1365,1785,1995,2145,2415,2618,2805,2926,3003,3045,3094,3135,
%T 3255,3315,3458,3542,3705,3795,3885,3910,3927,4186,4305,4370,4389,
%U 4466,4485,4515,4522,4641,4774,4785,4810,4845,4862,4930,4935,5005,5115,5187,5270,5278,5313,5330,5434,5474,5510,5565,5590
%N Products of 4 distinct primes numbers (or tetraprimes) that are deficient.
%F a(n) ~ A046390(n) ~ A046386(n) ~ A014613(n) ~ 6n log n / (log log n)^3. - _Charles R Greathouse IV_, Dec 06 2024
%e 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).
%e 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).
%t 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 *)
%o (PARI) catpr(~v, lim, mult, startAt)=forprime(p=startAt,lim\mult, listput(v,mult*p))
%o 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
%Y Intersection of A005100 and A046386. A046390 is a subsequence.
%Y Cf. A378480.
%K nonn
%O 1,1
%A _Massimo Kofler_, Dec 05 2024