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A372685 Prime numbers such that no lesser prime has the same binary weight (number of ones in binary expansion).

%I A372685 #23 May 13 2024 11:18:50
%S A372685 2,3,7,23,31,127,311,383,991,2039,3583,6143,8191,63487,73727,129023,
%T A372685 131071,522239,524287,1966079,4128767,14680063,16250879,33546239,
%U A372685 67108351,201064447,260046847,536739839,1073479679,2147483647,5335154687,8581545983,16911433727
%N A372685 Prime numbers such that no lesser prime has the same binary weight (number of ones in binary expansion).
%C A372685 The unsorted version is A061712.
%H A372685 Michael S. Branicky, <a href="/A372685/b372685.txt">Table of n, a(n) for n = 1..3320</a> (terms 36..3320 using A061712)
%F A372685 a(n) = prime(A372686(n)).
%e A372685 The terms together with their binary expansions and binary indices begin:
%e A372685      2:            10 ~ {2}
%e A372685      3:            11 ~ {1,2}
%e A372685      7:           111 ~ {1,2,3}
%e A372685     23:         10111 ~ {1,2,3,5}
%e A372685     31:         11111 ~ {1,2,3,4,5}
%e A372685    127:       1111111 ~ {1,2,3,4,5,6,7}
%e A372685    311:     100110111 ~ {1,2,3,5,6,9}
%e A372685    383:     101111111 ~ {1,2,3,4,5,6,7,9}
%e A372685    991:    1111011111 ~ {1,2,3,4,5,7,8,9,10}
%e A372685   2039:   11111110111 ~ {1,2,3,5,6,7,8,9,10,11}
%e A372685   3583:  110111111111 ~ {1,2,3,4,5,6,7,8,9,11,12}
%e A372685   6143: 1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
%t A372685 First/@GatherBy[Select[Range[1000],PrimeQ],DigitCount[#,2,1]&]
%o A372685 (Python)
%o A372685 from itertools import islice
%o A372685 from sympy import nextprime
%o A372685 def A372685_gen(): # generator of terms
%o A372685     p, a = 1, {}
%o A372685     while (p:=nextprime(p)):
%o A372685         if (c:=p.bit_count()) not in a:
%o A372685             yield p
%o A372685         a[c] = p
%o A372685 A372685_list = list(islice(A372685_gen(),20)) # _Chai Wah Wu_, May 12 2024
%Y A372685 This statistic (binary weight of primes) is A014499.
%Y A372685 Sorted version of A061712.
%Y A372685 For binary length instead of weight we have A104080, firsts of A035100.
%Y A372685 These primes have indices A372686, sorted version of A372517.
%Y A372685 A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A372685 A029837 gives greatest binary index, least A001511.
%Y A372685 A030190 gives binary expansion, reversed A030308.
%Y A372685 A035103 counts zeros in binary expansion of primes, firsts A372474.
%Y A372685 A048793 lists binary indices, reverse A272020, sum A029931.
%Y A372685 A372471 lists binary indices of primes.
%Y A372685 Cf. A000040, A005940, A066195, A069010, A070939, A071814, A211997, A372429, A372516, A372684.
%K A372685 nonn,base
%O A372685 1,1
%A A372685 _Gus Wiseman_, May 10 2024
%E A372685 a(22)-a(33) from _Chai Wah Wu_, May 12 2024