A052294 Pernicious numbers: numbers with a prime number of 1's in their binary expansion.

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 47, 48, 49, 50, 52, 55, 56, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 79, 80, 81, 82, 84, 87, 88, 91, 93, 94, 96, 97, 98, 100

Offset

1,1

Comments

No power of 2 is pernicious, but 2^n+1 always is.

If a prime p is of the form 2^k -1, then p is included in this sequence. - Leroy Quet, Sep 20 2008

There are A121497(n) n-bit members of this sequence. - Charles R Greathouse IV, Mar 22 2013

A list of programming codes for pernicious numbers can be found in the Rosetta Code link. - Martin Ettl, May 27 2014

Links

Iain Fox, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..4000 from Daniel Arribas)

Rosetta Code, Pernicious numbers

Example

26 is in the sequence because the binary expansion of 26 is 11010 and 11010 has three 1's and 3 is prime, so the number of 1's in the binary expansion of 26 is prime. - Omar E. Pol, Apr 04 2016

Maple

filter:= n -> isprime(convert(convert(n, base, 2), `+`)):
select(filter, [$1..1000]); # Robert Israel, Oct 19 2014

Mathematica

Select[Range[6! ], PrimeQ[DigitCount[ #, 2][[1]]]&] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)

Prog

(Haskell)
a052294 n = a052294_list !! (n-1)
a052294_list = filter ((== 1) . a010051 . a000120) [1..]
-- Reinhard Zumkeller, Nov 16 2012
(PARI) is(n)=isprime(hammingweight(n)) \\ Charles R Greathouse IV, Mar 22 2013
(Python)
from sympy import isprime
def ok(n): return isprime(bin(n).count("1"))
print([k for k in range(101) if ok(k)]) # Michael S. Branicky, Jun 16 2022
(Python)
from sympy import isprime
def ok(n): return isprime(n.bit_count())
print([k for k in range(101) if ok(k)]) # Michael S. Branicky, Dec 27 2023

Crossrefs

Cf. A000069, A001969, A010051, A000120, A081092 (primes).

Cf. A262481 (subsequence).

Sequence in context: A277727 A080943 A363690 * A336101 A267895 A216189

Adjacent sequences: A052291 A052292 A052293 * A052295 A052296 A052297

Keyword

easy,base,nice,nonn

Author

Jeremy Gow (jeremygo(AT)dai.ed.ac.uk), Feb 08 2000

Status

approved