login
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
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. Adams-Watters, Jul 28 2007
LINKS
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
Sequence in context: A168645 A282668 A117097 * A110615 A060679 A051234
KEYWORD
nonn
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. Adams-Watters. - Charles R Greathouse IV, Sep 24 2012
STATUS
approved