A372517 Least k such that the k-th prime number has exactly n ones in its binary expansion.

1, 2, 4, 9, 11, 64, 31, 76, 167, 309, 502, 801, 1028, 7281, 6363, 12079, 12251, 43237, 43390, 146605, 291640, 1046198, 951351, 2063216, 3957778, 11134645, 14198321, 28186247, 54387475, 249939829, 105097565, 393248783, 751545789, 1391572698, 2182112798, 8242984130

Offset

1,2

Comments

In other words, the a(n)-th prime is the least with binary weight n. The sorted version is A372686.

Links

Table of n, a(n) for n=1..36.

Formula

A000040(a(n)) = A061712(n).

Example

The primes A000040(a(n)) together with their binary expansions and binary indices begin:
        2:                     10 ~ {2}
        3:                     11 ~ {1,2}
        7:                    111 ~ {1,2,3}
       23:                  10111 ~ {1,2,3,5}
       31:                  11111 ~ {1,2,3,4,5}
      311:              100110111 ~ {1,2,3,5,6,9}
      127:                1111111 ~ {1,2,3,4,5,6,7}
      383:              101111111 ~ {1,2,3,4,5,6,7,9}
      991:             1111011111 ~ {1,2,3,4,5,7,8,9,10}
     2039:            11111110111 ~ {1,2,3,5,6,7,8,9,10,11}
     3583:           110111111111 ~ {1,2,3,4,5,6,7,8,9,11,12}
     6143:          1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
     8191:          1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
    73727:      10001111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,17}
    63487:       1111011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13,14,15,16}

Mathematica

spsm[y_]:=Max@@NestWhile[Most, y, Union[#]!=Range[Max@@#]&];
j=DigitCount[#, 2, 1]&/@Select[Range[1000], PrimeQ];
Table[Position[j, k][[1, 1]], {k, spsm[j]}]

Prog

(PARI) a(n) = my(k=1, p=2); while(hammingweight(p) !=n, p = nextprime(p+1); k++); k; \\ Michel Marcus, May 13 2024
(Python)
from itertools import count
from sympy import isprime, primepi
from sympy.utilities.iterables import multiset_permutations
def A372517(n):
    for l in count(n-1):
        m = 1<<l
        for d in multiset_permutations('0'*(l-n+1)+'1'*(n-1)):
            k = m+int('0'+''.join(d), 2)
            if isprime(k):
                return primepi(k) # Chai Wah Wu, May 13 2024

Crossrefs

Positions firsts of first appearances in A014499.

Taking primes gives A061712.

Counting zeros (weight) gives A372474, firsts of A035103.

For binary length we have A372684 (take primes A104080), firsts of A035100.

The sorted version is A372686, taking primes A372685.

A000120 counts ones in binary expansion (binary weight), zeros A080791.

A029837 gives greatest binary index, least A001511.

A030190 gives binary expansion, reversed A030308.

A048793 lists binary indices, reverse A272020, sum A029931.

A372471 lists binary indices of primes.

Cf. A000040, A066195, A069010, A070939, A071814, A211997, A372429, A372516.

Sequence in context: A292769 A307997 A372686 * A096134 A058885 A256446

Adjacent sequences: A372514 A372515 A372516 * A372518 A372519 A372520

Keyword

nonn,base

Author

Gus Wiseman, May 12 2024

Extensions

a(32)-a(36) from Pontus von Brömssen, May 13 2024

Status

approved