A168156 Sum of the binary digits of all primes between 2^(n-1) and 2^n-1, i.e., with exactly n binary digits.

0, 3, 5, 6, 18, 29, 56, 113, 240, 452, 885, 1790, 3474, 6951, 13671, 27183, 54201, 107224, 213882, 424513, 845716, 1682456, 3350362, 6671581, 13299828, 26500297, 52829961, 105342821, 210088965, 419106389, 836097752, 1668341390, 3329412989, 6645128078

Offset

1,2

Comments

Sequence A168155 yields the partial sums.

Links

Table of n, a(n) for n=1..34.

Example

No prime can be written with only 1 binary digit, thus a(1)=0.
The primes that can be written with 2 binary digits are 2 = 10[2] and 3 = 11[2], they have 3 nonzero bits, so a(2)=3.
Primes with 3 binary digits are 5 = 101[2] and 7 = 111[3]. They have a total of a(3)=5 nonzero bits.

Prog

(PARI) s=0; L=p=2; while( L*=2, print1(s", "); s=0; until( L<p=nextprime(p+1), s+=norml2(binary(p))))
(PARI) a(n)=my(s); forprime(p=2^(n-1), 2^n-1, s+=hammingweight(p)); s \\ Charles R Greathouse IV, Apr 07 2020

Crossrefs

Cf. A086904.

Sequence in context: A192119 A050563 A282809 * A295403 A272440 A276704

Adjacent sequences: A168153 A168154 A168155 * A168157 A168158 A168159

Keyword

nonn,base

Author

M. F. Hasler, Nov 20 2009

Extensions

a(26)-a(32) from Donovan Johnson, Jul 28 2010

a(33) from Chai Wah Wu, Apr 06 2020

a(34) from Chai Wah Wu, Apr 07 2020

Status

approved