login
Primes in whose binary expansion the number of 0-bits is one more than the number of 1-bits.
7

%I #34 Aug 29 2024 01:19:37

%S 17,67,73,97,263,269,277,281,293,337,353,389,401,449,1039,1051,1063,

%T 1069,1109,1123,1129,1163,1171,1187,1193,1201,1249,1291,1301,1321,

%U 1361,1543,1549,1571,1609,1667,1669,1697,1801,4127,4157,4211,4217

%N Primes in whose binary expansion the number of 0-bits is one more than the number of 1-bits.

%C A010051(a(n)) = 1 and A037861(a(n)) = 1. - _Reinhard Zumkeller_, Mar 31 2015

%H Indranil Ghosh, <a href="/A095072/b095072.txt">Table of n, a(n) for n = 1..20000</a> (terms 1..1000 from Reinhard Zumkeller)

%H A. Karttunen and J. Moyer, <a href="/A095062/a095062.c.txt">C-program for computing the initial terms of this sequence</a>

%e 97 is in the sequence because 97 is a prime and 97_10 = 1100001_2. The number of 0's in 1100001 is 4 and the number of 1's is 3. - _Indranil Ghosh_, Jan 31 2017

%t Select[Prime[Range[500]], Differences[DigitCount[#, 2]] == {1} &]

%o (PARI) isA095072(n)=my(v=binary(n));#v==2*sum(i=1,#v,v[i])+1&&isprime(n)

%o (PARI) forprime(p=2, 4250, v=binary(p); s=0; for(k=1, #v, s+=if(v[k]==0,+1,-1)); if(s==1,print1(p,", ")))

%o (Haskell)

%o a095072 n = a095072_list !! (n-1)

%o a095072_list = filter ((== 1) . a010051' . fromIntegral) a031444_list

%o -- _Reinhard Zumkeller_, Mar 31 2015

%o (Python)

%o #Program to generate the b-file

%o from sympy import isprime

%o i=1

%o j=1

%o while j<=200:

%o if isprime(i) and bin(i)[2:].count("0")-bin(i)[2:].count("1")==1:

%o print(str(j)+" "+str(i))

%o j+=1

%o i+=1 # _Indranil Ghosh_, Jan 31 2017

%Y Intersection of A000040 and A031444. Subset of A095071.

%Y Cf. A095052.

%Y Cf. A010051, A037861.

%K nonn,base,easy

%O 1,1

%A _Antti Karttunen_, Jun 01 2004