cp's OEIS Frontend

A077135 Composite numbers n whose proper (other than 1 and n) odd divisors are prime and even divisors are 1 less than a prime.

%I A077135 #19 Apr 05 2019 10:19:59
%S A077135 4,6,8,9,10,12,14,15,20,21,22,25,26,33,34,35,38,39,44,46,49,51,55,57,
%T A077135 58,62,65,69,74,77,82,85,86,87,91,92,93,94,95,106,111,115,116,118,119,
%U A077135 121,122,123,129,133,134,141,142,143,145,146,155,158,159,161,164,166,169
%N A077135 Composite numbers n whose proper (other than 1 and n) odd divisors are prime and even divisors are 1 less than a prime.
%C A077135 k is a member if (1) k = p*q p, q are primes; (2) k = 4*p and p, 2p+1 are primes. Are there any other prime signatures k could take?
%C A077135 The odd members (A046315) outnumber the even members. - _Robert G. Wilson v_, Mar 31 2005
%C A077135 This sequence consists of precisely the semiprimes and numbers of the form 4p where 2p+1 is also prime. n cannot have pq as a proper divisor, with p and q odd primes (not necessarily distinct). Likewise 8 cannot be a proper factor. This eliminates all but the specified cases. - _Franklin T. Adams-Watters_, Jul 28 2007
%H A077135 Vincenzo Librandi, <a href="/A077135/b077135.txt">Table of n, a(n) for n = 1..7000</a>
%t A077135 fQ[n_] := Block[{d = Take[ Divisors[n], {2, -2}]}, Union[ Flatten[ PrimeQ[{Select[d, OddQ[ # ] &], Select[d, EvenQ[ # ] &] + 1}]]] == {True}]; Select[ Range[ 176], fQ[ # ] &] (* _Robert G. Wilson v_, Mar 31 2005 *)
%t A077135 cnQ[n_]:=Module[{d=Most[Rest[Divisors[n]]]},AllTrue[Join[Select[ d,OddQ],Select[ d,EvenQ]+1],PrimeQ]]; Select[Range[200],CompositeQ[#] && cnQ[#]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Apr 05 2019 *)
%o A077135 (PARI) is(n)=fordiv(n,d,if(!isprime(bitor(d,1)) && d>1, return(d==n))); !isprime(n) && n>1 \\ _Charles R Greathouse IV_, Sep 24 2012
%Y A077135 Cf. A001358, A005384.
%K A077135 nonn
%O A077135 1,1
%A A077135 _Amarnath Murthy_, Oct 29 2002
%E A077135 Corrected and extended by _Robert G. Wilson v_, Mar 31 2005
%E A077135 Definition corrected, following an observation by _Franklin T. Adams-Watters_. - _Charles R Greathouse IV_, Sep 24 2012