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

Author

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

Keywords

• easy
• base
• nice
• nonn

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

Examples

			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
		

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

Crossrefs

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

Cf. A262481 (subsequence).

Programs

Haskell

a052294 n = a052294_list !! (n-1)
a052294_list = filter ((== 1) . a010051 . a000120) [1..]
-- Reinhard Zumkeller, Nov 16 2012

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 *)

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