A014499 Number of 1's in binary representation of n-th prime.

1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 3, 3, 4, 5, 4, 5, 5, 3, 4, 3, 5, 4, 4, 3, 4, 5, 5, 5, 4, 7, 3, 3, 4, 4, 5, 5, 4, 5, 5, 5, 5, 7, 3, 4, 5, 5, 7, 5, 5, 5, 7, 5, 7, 2, 4, 4, 5, 4, 4, 5, 4, 5, 6, 5, 6, 5, 4, 6, 6, 4, 6, 7, 6, 7, 8, 4, 5, 4, 5, 5, 5, 7, 5, 7, 7, 4, 5, 6, 7, 6, 8, 7, 7, 7, 8, 8, 3, 4

Offset: 1

Author

Ingemar Assarsjo (ingemar(AT)binomen.se)

Keywords

• nonn
• base
• easy

Comments

a(n) is the rank of prime(n) in the base-2 dominance order on the natural numbers. - Tom Edgar, Mar 25 2014

Examples

			From _M. F. Hasler_, Mar 03 2023: (Start)
a(n) = 1 only for p(n = 1) = 2, the only prime equal to a power of 2.
a(n) = 2 for n in A159611 = A000720(A019434) = {2, 3, 7, 55, 6543} (probably complete), the Fermat primes F[k] = 2^2^k + 1 with k = 0, 1, 2, 3, 4. (On the graph one can distinctly see a(6543) = 2 corresponding to F[4] = 65537.)
a(n) = 3 for n in A000720(A081091) = (4, 5, 6, 8, 12, 13, 19, 21, 25, 32, 33, 44, 98, 106, 116, 136, 174, 191, 310, 313, 319, 565, 568, ...). (End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
• Christian Elsholtz, Almost all primes have a multiple of small Hamming weight, arXiv:1602.05974 [math.NT], 2016.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A035103, A035100, A004676, A090455.

Cf. A027697, A027699

Cf. A180024. - Reinhard Zumkeller, Aug 08 2010

Cf. A072084.

Cf. A159611 (indices of 2s), A000720(A081091) (indices of 3s). - M. F. Hasler, Mar 03 2023

Programs

Haskell

a014499 = a000120 . a000040  -- Reinhard Zumkeller, Feb 10 2013

Magma

[&+Intseq(NthPrime(n), 2): n in [1..100] ]; // Vincenzo Librandi, Mar 25 2014

Mathematica

Table[Plus @@ IntegerDigits[Prime[n], 2], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

PARI

A014499(n)=hammingweight(prime(n)) \\ M. F. Hasler, Nov 20 2009, updated Mar 03 2023

Python

from sympy import prime
def A014499(n): return prime(n).bit_count() # Chai Wah Wu, Mar 22 2023

Sage

[sum(i.digits(base=2)) for i in primes_first_n(200)] # Tom Edgar, Mar 25 2014

Formula

a(n) = A000120(A000040(n)).

a(A049084(A061712(n))) = n. - Reinhard Zumkeller, Feb 10 2013

a(n) = [x^prime(n)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018