A121497 -id:A121497 - cp's OEIS frontend

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

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

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

			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

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

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

Cf. A262481 (subsequence).

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

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

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

is(n)=isprime(hammingweight(n)) \\ Charles R Greathouse IV, Mar 22 2013

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

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

A052467 Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.

Original entry on oeis.org

0, 1, 3, 6, 11, 20, 37, 70, 134, 255, 476, 869, 1564, 2821, 5201, 9948, 19793, 40562, 84271, 174952, 359576, 728805, 1457402, 2885051, 5681277, 11185110, 22103926, 43939533, 87864092, 176447165, 354929146, 713198803, 1428312446, 2846268351

Offset: 1

Eric W. Weisstein

nonn

Number of compositions of n into a prime number of parts. - Vladeta Jovovic, Jan 31 2005

The number of pernicious numbers (A052294) between 2^(n-1) and 2^n. Although the graph looks almost like 2^n, the graph of a(n)/2^n has quite a bit of variation. - T. D. Noe, Mar 14 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Binomial Transform.

Cf. A038499.

Cf. A010051, A121497.

b[n_] := Boole[ PrimeQ[n]]; a[n_] := Sum[ Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 34}] // Differences (* Jean-François Alcover, Oct 25 2012 *)

G.f.: Sum_{k>=1} (x/(1 - x))^prime(k). - Ilya Gutkovskiy, Dec 28 2016

a(n) = A121497(n+1) - A121497(n). - Wesley Ivan Hurt, Jan 14 2022

More terms from David Wasserman, Feb 25 2002

Description corrected by T. D. Noe, May 17 2003

A178851 The number of length n sequences on {0,1,2}(ternary sequences) that contain a prime number of 2's.

Original entry on oeis.org

0, 0, 1, 7, 32, 121, 412, 1317, 4048, 12144, 35904, 105249, 306968, 892217, 2585468, 7468532, 21500800, 61688513, 176477988, 503906221, 1438235592, 4110846808, 11789919200, 33991337521, 98657320240, 288505634480, 850146795840, 2522918119392, 7531922736384

Offset: 0

Geoffrey Critzer, Dec 27 2010

nonn

a(n) is the number of positive integers less than 3^n that when expressed as a ternary numeral contain a prime number of 2's.

a(n)/3^n is the probability that a series of Bernoulli trials with probability of success equal to 1/3 will result in a prime number of successes. Cf. A121497

			a(3)=7 because 8,17,20,23,24,25,26 have a prime number of 2's in their ternary notation.

Eric M. Schmidt, Table of n, a(n) for n = 0..2000

P=Table[Prime[m],{m,1,200}];Range[0,20]! CoefficientList[Series[Exp[2x] Sum[x^p/p!,{p,P}],{x,0,20}],x]

E.g.f.:exp(2x)*(x^2/2!+x^3/3!+x^5/5!+...)

a(n) = Sum Binomial(n,p)*2^(n-p) where the sum is taken over the prime numbers.

A334038 a(n) = Product_{p<=n, p prime} binomial(n,p).

Original entry on oeis.org

1, 1, 1, 3, 24, 100, 1800, 15435, 702464, 13716864, 163296000, 1383574500, 109294479360, 3842829083808, 1159801183597056, 132320316074821875, 8213884352593920000, 327816138093181337600, 167079259535068179726336, 34044607357920579594754944

Offset: 0

Om R. Patel, Apr 13 2020

nonn

			For n=2, p=2: a(n) = C(2,2) = 1.
For n=3, p=2,3: a(n) = C(3,2) * C(3,3) = 3.
For n=4, p=2,3: a(n) = C(4,2) * C(4,3) = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..145

Cf. A000040, A001142, A010051, A121497.

a:= n-> mul(`if`(isprime(p), binomial(n, p), 1), p=2..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 13 2020

a(n) = prod(k=1, n, if (isprime(k), binomial(n, k), 1)); \\ Michel Marcus, Apr 13 2020

a(n)=my(s); forprime(p=2,n, s*=binomial(n,p)); s \\ Charles R Greathouse IV, Apr 13 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

A052467 Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A178851 The number of length n sequences on {0,1,2}(ternary sequences) that contain a prime number of 2's.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

Formula

A334038 a(n) = Product_{p<=n, p prime} binomial(n,p).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

PARI