

A077135


Composite numbers n whose proper (other than 1 and n) odd divisors are prime and even divisors are 1 less than a prime.


2



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

1,1


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 odd members (A046315) outnumber the even members.  Robert G. Wilson v, Mar 31 2005
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. AdamsWatters, Jul 28 2007


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..7000


MATHEMATICA

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 *)


PROG

(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


CROSSREFS

Cf. A001358, A005384.
Sequence in context: A168645 A282668 A117097 * A110615 A060679 A051234
Adjacent sequences: A077132 A077133 A077134 * A077136 A077137 A077138


KEYWORD

nonn,changed


AUTHOR

Amarnath Murthy, Oct 29 2002


EXTENSIONS

Corrected and extended by Robert G. Wilson v, Mar 31 2005
Definition corrected, following an observation by Franklin T. AdamsWatters.  Charles R Greathouse IV, Sep 24 2012


STATUS

approved



