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A215173 Numbers k such that k and k+1 are both of the form p*q^3 where p and q are distinct primes.

%I A215173 #14 Nov 22 2020 03:06:55
%S A215173 135,296,375,1431,1592,3992,4023,6183,7624,8936,9368,10071,10232,
%T A215173 10375,10984,13256,16551,16712,19143,20871,22328,22375,23031,24488,
%U A215173 28375,28376,28647,33271,34856,35127,40311,40472,41336,43767,46791,49624,50408,52375,53271
%N A215173 Numbers k such that k and k+1 are both of the form p*q^3 where p and q are distinct primes.
%C A215173 Intersection of A065036 and A065036 - 1. - _Robert Israel_, Jun 15 2014
%H A215173 Amiram Eldar, <a href="/A215173/b215173.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2048 from Robert Israel)
%e A215173 135 is a member as 135 = 5*3^3 and 136 = 17*2^3.
%p A215173 with(numtheory):for n from 1 to 55000 do:x:=factorset(n):y:=factorset(n+1):x2:=sqrt(n):y2:=sqrt(n+1):n1:=nops(x):n2:=nops(y):if n1=2 and n2=2 and bigomega(n) = 4 and bigomega(n+1) = 4 and x2<>floor(x2) and y2<>floor(y2) then printf("%a, ", n):else fi:od:
%p A215173 # Alternative:
%p A215173 N:= 10^5: # to get all terms < N
%p A215173 P1:= select(isprime,{2,seq(2*i+1,i=1..floor(N/16))}):
%p A215173 P2:= select(t -> t^3 <= N/2,P1):
%p A215173 B:= {seq(seq(p^3*q,q=select(`<`,P1,floor(N/p^3)) minus {p}),p=P2)}:
%p A215173 B intersect map(`-`,B,1); # _Robert Israel_, Jun 15 2014
%t A215173 lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 3}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 3}, AppendTo[lst, n]]], {n, 3, 55000}]; lst
%Y A215173 Cf. A074172, A065036.
%K A215173 nonn
%O A215173 1,1
%A A215173 _Michel Lagneau_, Aug 05 2012