reviewed

approved

proposed

reviewed

editing

proposed

seqQ[n_] := CompositeQ[n] && AllTrue[Most @ Rest @ Divisors[n], PrimeQ[#] || PrimeQ[# - 1] &]; Select[Range[161], seqQ] (* Amiram Eldar, Dec 10 2019 *)

Amiram Eldar, <a href="/A077136/b077136.txt">Table of n, a(n) for n = 1..10000</a>

approved

editing

_Amarnath Murthy (amarnath_murthy(AT)yahoo.com), _, Oct 29 2002

Corrected and extended by _Ralf Stephan (ralf(AT)ark.in-berlin.de), _, Mar 23 2003

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

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

nonn,new

nonn

(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", ")))

nonn,new

nonn