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%I #13 Dec 10 2019 04:00:37
%S 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 57,58,62,65,69,74,76,77,82,85,86,87,91,93,94,95,106,111,115,118,119,
%U 121,122,123,124,129,133,134,141,142,143,145,146,148,155,158,159,161
%N Composite numbers n whose proper divisors (excluding 1 and n) are all of the form p or p+1, with p prime.
%C 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 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_, Jul 28 2007
%H Amiram Eldar, <a href="/A077136/b077136.txt">Table of n, a(n) for n = 1..10000</a>
%t seqQ[n_] := CompositeQ[n] && AllTrue[Most @ Rest @ Divisors[n], PrimeQ[#] || PrimeQ[# - 1] &]; Select[Range[161], seqQ] (* _Amiram Eldar_, Dec 10 2019 *)
%o (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 Cf. A077135.
%K nonn
%O 1,1
%A _Amarnath Murthy_, Oct 29 2002
%E Corrected and extended by _Ralf Stephan_, Mar 23 2003