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.
In the range ]2^4,2^5] 17 (10001 in binary) is the only such prime thus a(2) = 1.
a031444 n = a031444_list !! (n-1) a031444_list = filter ((== 1) . a037861) [1..] -- Reinhard Zumkeller, Mar 31 2015
Select[Range[350], (Differences@ DigitCount[#, 2])[[1]] == 1 &] (* Amiram Eldar, Aug 03 2023 *)
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 *)
73 is in the sequence because 73 is a prime and 73_10 = 1001001_2. '1001001' has four 0's and one 1. - _Indranil Ghosh_, Jan 31 2017
Reap[Do[p=Prime[k];id=IntegerDigits[p,2];n=Length@id;If[Count[id,0]>n/2,Sow[p]],{k,200}]][[2,1]]
(* Zak Seidov *)
Select[Prime[Range[200]],DigitCount[#,2,0]>DigitCount[#,2,1]&] (* Harvey P. Dale, Nov 28 2024 *)
B(x) = { nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b0 > b1, return(1);, return(0););};
forprime(x = 2, 1171, if(B(x), print1(x, ", "); ); ); \\ Washington Bomfim, Jan 11 2011
{forprime(p=2,1171,nB=floor(log(p)/log(2));
sum(i=0,nB,bittest(p,i))<=nB/2&print1(p,","))} \\ Zak Seidov, Jan 11 2011
#Program to generate the b-file
from sympy import isprime
i=1
j=1
while j<=200:
if isprime(i) and bin(i)[2:].count("0")>bin(i)[2:].count("1"):
print(str(j)+" "+str(i))
j+=1
i+=1 # Indranil Ghosh, Jan 31 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&]
The binary expansion of 17 is (1,0,0,0,1) with ones minus zeros 2 - 3 = -1, and 17 is the 7th prime, 7 is in the sequence.
The primes A000040(a(n)) together with their binary expansions and binary indices begin:
17: 10001 ~ {1,5}
67: 1000011 ~ {1,2,7}
73: 1001001 ~ {1,4,7}
97: 1100001 ~ {1,6,7}
263: 100000111 ~ {1,2,3,9}
269: 100001101 ~ {1,3,4,9}
277: 100010101 ~ {1,3,5,9}
281: 100011001 ~ {1,4,5,9}
293: 100100101 ~ {1,3,6,9}
337: 101010001 ~ {1,5,7,9}
353: 101100001 ~ {1,6,7,9}
389: 110000101 ~ {1,3,8,9}
401: 110010001 ~ {1,5,8,9}
449: 111000001 ~ {1,7,8,9}
1039: 10000001111 ~ {1,2,3,4,11}
1051: 10000011011 ~ {1,2,4,5,11}
1063: 10000100111 ~ {1,2,3,6,11}
1069: 10000101101 ~ {1,3,4,6,11}
1109: 10001010101 ~ {1,3,5,7,11}
1123: 10001100011 ~ {1,2,6,7,11}
1129: 10001101001 ~ {1,4,6,7,11}
1163: 10010001011 ~ {1,2,4,8,11}
Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==-1&]
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