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A077135
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Composite numbers n whose proper (other than 1 and n) odd divisors are prime and even divisors are 1 less than a prime.
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2
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4, 6, 8, 9, 10, 12, 14, 15, 20, 21, 22, 25, 26, 33, 34, 35, 38, 39, 44, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 106, 111, 115, 116, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 164, 166, 169
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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?
This sequence consists of precisely the semiprimes and numbers of the form 4p where 2p+1 is also prime. n cannot have pq as a proper divisor, with p and q odd primes (not necessarily distinct). Likewise 8 cannot be a proper factor. This eliminates all but the specified cases. - Franklin T. Adams-Watters, Jul 28 2007
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LINKS
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MATHEMATICA
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fQ[n_] := Block[{d = Take[ Divisors[n], {2, -2}]}, Union[ Flatten[ PrimeQ[{Select[d, OddQ[ # ] &], Select[d, EvenQ[ # ] &] + 1}]]] == {True}]; Select[ Range[ 176], fQ[ # ] &] (* Robert G. Wilson v, Mar 31 2005 *)
cnQ[n_]:=Module[{d=Most[Rest[Divisors[n]]]}, AllTrue[Join[Select[ d, OddQ], Select[ d, EvenQ]+1], PrimeQ]]; Select[Range[200], CompositeQ[#] && cnQ[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 05 2019 *)
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PROG
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(PARI) is(n)=fordiv(n, d, if(!isprime(bitor(d, 1)) && d>1, return(d==n))); !isprime(n) && n>1 \\ Charles R Greathouse IV, Sep 24 2012
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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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