A077136 Composite numbers n whose proper divisors (excluding 1 and n) are all of the form p or p+1, with p prime.

4, 6, 8, 9, 10, 12, 14, 15, 16, 21, 22, 24, 25, 26, 28, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 76, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 124, 129, 133, 134, 141, 142, 143, 145, 146, 148, 155, 158, 159, 161

Offset: 1

Author

Amarnath Murthy, Oct 29 2002

Keywords

• nonn

Comments

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?

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A077135.

Programs

Mathematica

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

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",")))

Extensions

Corrected and extended by Ralf Stephan, Mar 23 2003