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A077136
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Composite numbers n whose proper divisors (excluding 1 and n) are all of the form p or p+1, with p prime.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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MATHEMATICA
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seqQ[n_] := CompositeQ[n] && AllTrue[Most @ Rest @ Divisors[n], PrimeQ[#] || PrimeQ[# - 1] &]; Select[Range[161], seqQ] (* Amiram Eldar, Dec 10 2019 *)
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PROG
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(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", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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