This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The binary expansion of 83 is (1,0,1,0,0,1,1), and 83 is the 23rd prime, so a(23) = 4 - 3 = 1.
Table[DigitCount[Prime[n],2,1]-DigitCount[Prime[n],2,0],{n,100}]
DigitCount[#,2,1]-DigitCount[#,2,0]&/@Prime[Range[100]] (* Harvey P. Dale, May 09 2025 *)
From _Indranil Ghosh_, Feb 03 2017: (Start) 17 is in the sequence because 17_10 = 10001_2. '10001' has two 1's and three 0's. 37 is in the sequence because 37_10 = 100101_2. '100101' has three 1's and 3 0's. (End)
Select[Prime[Range[150]], Differences[DigitCount[#, 2]][[1]] >= 0 &] (* Amiram Eldar, Jul 25 2023 *) Select[Prime[Range[150]],DigitCount[#,2,1]<=DigitCount[#,2,0]&] (* Harvey P. Dale, Sep 27 2023 *)
B(x) = {nB = floor(log(x)/log(2)); z1 = 0; z0 = 0;
for(i = 0, nB, if(bittest(x,i), z1++;, z0++;); );
if(z1 <= z0, return(1);, return(0););};
forprime(x = 2, 787, if(B(x), print1(x, ", "); ); );
\\ Washington Bomfim, Jan 11 2011
from sympy import isprime
i=1
j=1
while j<=250:
if isprime(i) and bin(i)[2:].count("1")<=bin(i)[2:].count("0"):
print(str(j)+" "+str(i))
j+=1
i+=1 # Indranil Ghosh, Feb 03 2017
71 is in the sequence because 71_10 = 1000111_2. '1000111' has four 1's and three 0's. - _Indranil Ghosh_, Feb 03 2017
Select[Prime[Range[500]], Differences[DigitCount[#, 2]] == {-1} &]
{ forprime(p=2, 2000,
v=binary(p); s=0;
for(k=1,#v, s+=if(v[k]==1,+1,-1));
if(s==1,print1(p,", "))
) }
from sympy import isprime
i=1
j=1
while j<=25000:
if isprime(i) and bin(i)[2:].count("1")-bin(i)[2:].count("0")==1:
print(str(j)+" "+str(i))
j+=1
i+=1 # Indranil Ghosh, Feb 03 2017
13 is in the sequence because 13 is prime and 13 = 1101_2. '1101' has three 1's and one 0. 3 > 1 + 1. - _Indranil Ghosh_, Feb 07 2017
B(x) = {nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b1 > (b0+1), return(1);, return(0);); };
forprime(x = 3, 499, if(B(x), print1(x, ", "); ); );
\\ Washington Bomfim, Jan 11 2011
from sympy import isprime
i = 1
j = 1
while j <= 2000:
bi = bin(i)[2:]
if isprime(i) and bi.count("1") > 1 + bi.count("0"):
print(str(j) + " " + str(i))
j += 1
i += 1 # Indranil Ghosh, Feb 07 2017
The binary expansion of 83 is (1,0,1,0,0,1,1) with ones minus zeros 4 - 3 = 1, and 83 is the 23rd prime, so 23 is in the sequence.
The primes A000040(a(n)) together with their binary expansions and binary indices begin:
5: 101 ~ {1,3}
19: 10011 ~ {1,2,5}
71: 1000111 ~ {1,2,3,7}
83: 1010011 ~ {1,2,5,7}
89: 1011001 ~ {1,4,5,7}
101: 1100101 ~ {1,3,6,7}
113: 1110001 ~ {1,5,6,7}
271: 100001111 ~ {1,2,3,4,9}
283: 100011011 ~ {1,2,4,5,9}
307: 100110011 ~ {1,2,5,6,9}
313: 100111001 ~ {1,4,5,6,9}
331: 101001011 ~ {1,2,4,7,9}
397: 110001101 ~ {1,3,4,8,9}
409: 110011001 ~ {1,4,5,8,9}
419: 110100011 ~ {1,2,6,8,9}
421: 110100101 ~ {1,3,6,8,9}
433: 110110001 ~ {1,5,6,8,9}
457: 111001001 ~ {1,4,7,8,9}
1103: 10001001111 ~ {1,2,3,4,7,11}
1117: 10001011101 ~ {1,3,4,5,7,11}
1181: 10010011101 ~ {1,3,4,5,8,11}
1223: 10011000111 ~ {1,2,3,7,8,11}
Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==1&]
okQ[n_] := Module[{b = IntegerDigits[n, 2]}, Count[b, 1] > 2*Count[b, 0]]; Select[Prime[Range[200]], okQ] (* T. D. Noe, May 02 2013 *)
has(n)=3*hammingweight(n)>2*#binary(n) select(has,primes(500))
q:= proc(n) local l, t, i; l:= Bits[Split](n); t:=0;
for i to nops(l) do t:= t-1+2*l[i];
if t<0 then return false fi
od: true
end:
select(isprime and q, [$2..500])[]; # Alois P. Heinz, Jan 07 2022
q[n_] := PrimeQ[n] && AllTrue[Accumulate[(-1)^Reverse[IntegerDigits[n, 2]]], # <= 0 &]; Select[Range[500], q] (* Amiram Eldar, Jan 07 2022 *)
from sympy import isprime
def ok(n):
if n == 0: return True
count = {"0": 0, "1": 0}
for bit in bin(n)[:1:-1]:
count[bit] += 1
if count["0"] > count["1"]: return False
return isprime(n)
print([k for k in range(3, 500, 2) if ok(k)]) # Michael S. Branicky, Jan 07 2022
71 is in the sequence because 71 is a prime and 71_10 = 1000111_2. '1000111' has four 1's and three 0's.
Select[Range[400], PrimeQ[#] && First[d = DigitCount[#, 2]] > Last[d] > 0 &] (* Amiram Eldar, Jan 08 2021 *)
isok(n) = if (isprime(n), my(nb=#binary(n), h=hammingweight(n)); (2*h > nb) && (h < nb)); \\ Michel Marcus, Jan 10 2021
from sympy import sieve A340466_list = [p for p in sieve.primerange(1,10**4) if len(bin(p))-2 < 2*bin(p).count('1') < 2*len(bin(p))-4] # Chai Wah Wu, Jan 10 2021
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