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

%I A372517 #20 May 13 2024 12:22:44
%S A372517 1,2,4,9,11,64,31,76,167,309,502,801,1028,7281,6363,12079,12251,43237,
%T A372517 43390,146605,291640,1046198,951351,2063216,3957778,11134645,14198321,
%U A372517 28186247,54387475,249939829,105097565,393248783,751545789,1391572698,2182112798,8242984130
%N A372517 Least k such that the k-th prime number has exactly n ones in its binary expansion.
%C A372517 In other words, the a(n)-th prime is the least with binary weight n. The sorted version is A372686.
%F A372517 A000040(a(n)) = A061712(n).
%e A372517 The primes A000040(a(n)) together with their binary expansions and binary indices begin:
%e A372517         2:                     10 ~ {2}
%e A372517         3:                     11 ~ {1,2}
%e A372517         7:                    111 ~ {1,2,3}
%e A372517        23:                  10111 ~ {1,2,3,5}
%e A372517        31:                  11111 ~ {1,2,3,4,5}
%e A372517       311:              100110111 ~ {1,2,3,5,6,9}
%e A372517       127:                1111111 ~ {1,2,3,4,5,6,7}
%e A372517       383:              101111111 ~ {1,2,3,4,5,6,7,9}
%e A372517       991:             1111011111 ~ {1,2,3,4,5,7,8,9,10}
%e A372517      2039:            11111110111 ~ {1,2,3,5,6,7,8,9,10,11}
%e A372517      3583:           110111111111 ~ {1,2,3,4,5,6,7,8,9,11,12}
%e A372517      6143:          1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
%e A372517      8191:          1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
%e A372517     73727:      10001111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,17}
%e A372517     63487:       1111011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13,14,15,16}
%t A372517 spsm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
%t A372517 j=DigitCount[#,2,1]&/@Select[Range[1000],PrimeQ];
%t A372517 Table[Position[j,k][[1,1]],{k,spsm[j]}]
%o A372517 (PARI) a(n) = my(k=1, p=2); while(hammingweight(p) !=n, p = nextprime(p+1); k++); k; \\ _Michel Marcus_, May 13 2024
%o A372517 (Python)
%o A372517 from itertools import count
%o A372517 from sympy import isprime, primepi
%o A372517 from sympy.utilities.iterables import multiset_permutations
%o A372517 def A372517(n):
%o A372517     for l in count(n-1):
%o A372517         m = 1<<l
%o A372517         for d in multiset_permutations('0'*(l-n+1)+'1'*(n-1)):
%o A372517             k = m+int('0'+''.join(d),2)
%o A372517             if isprime(k):
%o A372517                 return primepi(k) # _Chai Wah Wu_, May 13 2024
%Y A372517 Positions firsts of first appearances in A014499.
%Y A372517 Taking primes gives A061712.
%Y A372517 Counting zeros (weight) gives A372474, firsts of A035103.
%Y A372517 For binary length we have A372684 (take primes A104080), firsts of A035100.
%Y A372517 The sorted version is A372686, taking primes A372685.
%Y A372517 A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A372517 A029837 gives greatest binary index, least A001511.
%Y A372517 A030190 gives binary expansion, reversed A030308.
%Y A372517 A048793 lists binary indices, reverse A272020, sum A029931.
%Y A372517 A372471 lists binary indices of primes.
%Y A372517 Cf. A000040, A066195, A069010, A070939, A071814, A211997, A372429, A372516.
%K A372517 nonn,base
%O A372517 1,2
%A A372517 _Gus Wiseman_, May 12 2024
%E A372517 a(32)-a(36) from _Pontus von Brömssen_, May 13 2024