%I A077136
%S A077136 4,6,8,9,10,12,14,15,16,21,22,24,25,26,28,33,34,35,38,39,46,49,51,55,
%T A077136 57,58,62,65,69,74,76,77,82,85,86,87,91,93,94,95,106,111,115,118,119,
%U A077136 121,122,123,124,129,133,134,141,142,143,145,146,148,155,158,159,161
%N A077136 Composite numbers n whose proper divisors (excluding 1 and n) are all 
               of the form p or p+1, with p prime.
%C A077136 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 A077136 The only numbers in the sequence that are neither a semiprime nor of 
               the form 4p (where 2p-1 is also prime) are 16 and 24. If n has pq 
               as a proper divisor, with p and q odd primes (not necessarily distinct), 
               neither pq nor pq-1 can be prime. Likewise 16 cannot be a proper 
               factor. Other than the two specified cases, this leaves n = 8p, where 
               2p-1 and 4p-1 are primes. p = 2 or 3 gives the exceptional cases 
               16 and 24, respectively. Any other prime must be == 1 or 2 (mod 3); 
               if 1, then 4p-1 is divisible by 3 and if 2, then 2p-1 is divisible 
               by 3. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jul 
               28 2007
%o A077136 (PARI) for(n=1,200,v=divisors(n):s=0:for(k=2,length(v)-1,if(isprime(v[k])||isprime(v[k]-1),
               s=s+1)): if(s&&s==length(v)-2,print1(n",")))
%Y A077136 Cf. A077135.
%Y A077136 Sequence in context: A163122 A050655 A117098 this_sequence A098216 A063806 
               A063989
%Y A077136 Adjacent sequences: A077133 A077134 A077135 this_sequence A077137 A077138 
               A077139
%K A077136 nonn
%O A077136 1,1
%A A077136 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 29 2002
%E A077136 Corrected and extended by Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 
               23 2003