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A308736 Numbers n such that n, n+2, n+4, n+6 are of the form p^2*q where p and q are distinct primes.

%I A308736 #33 Jan 03 2022 19:25:25
%S A308736 2523,3112819,5656019,10132171,12167825,16639567,25302173,31995475,
%T A308736 35158921,37334419,43890719,44816821,47715269,53548223,55534523,
%U A308736 90526075,90533525,127558319,142929025,143167073,144989575,147182225
%N A308736 Numbers n such that n, n+2, n+4, n+6 are of the form p^2*q where p and q are distinct primes.
%C A308736 All terms are odd. Proof: if n is even then out of the 4 numbers n, n+2, n+4, n+6, 2 of them must be either both of the form 2*p^2, 2*q^2, or both of the form 4*p, 4*q. In either case, for p != q and p, q prime, the difference between these 2 numbers are more than 6, reaching a contradiction. - _Chai Wah Wu_, Jun 24 2019
%H A308736 Giovanni Resta, <a href="/A308736/b308736.txt">Table of n, a(n) for n = 1..10000</a> (first 123 terms from Ray Chandler)
%e A308736 2523 = 3*29*29, 2525 = 5*5*101, 2527 = 7*19*19, 2529 = 3*3*281.
%t A308736 psx = Table[{0}, {7}]; nmax = 150000000; n = 1; lst = {};
%t A308736 While[n < nmax, n++;
%t A308736   psx = RotateRight[psx];
%t A308736   psx[[1]] = Sort[Last /@ FactorInteger[n]];
%t A308736   If[Union[{psx[[1]], psx[[3]], psx[[5]], psx[[7]]}] == {{1, 2}}, AppendTo[lst, n - 6]];];
%t A308736 lst
%o A308736 (Python)
%o A308736 from sympy import factorint
%o A308736 A308736_list, n, mlist = [], 3, [False]*4
%o A308736 while len(A308736_list) < 100:
%o A308736     if mlist[0] and mlist[1] and mlist[2] and mlist[3]:
%o A308736         A308736_list.append(n)
%o A308736     n += 2
%o A308736     f = factorint(n+6)
%o A308736     mlist = mlist[1:] + [(len(f),sum(f.values())) == (2,3)] # _Chai Wah Wu_, Jun 24 2019, Jan 03 2022.
%Y A308736 Cf. A074173, A308735.
%K A308736 nonn
%O A308736 1,1
%A A308736 _Ray Chandler_, Jun 24 2019