cp's OEIS Frontend

A308430 Number of 0's minus number of 1's among the edge truncated binary representations of the first n prime numbers.

%I A308430 #38 Dec 27 2024 18:30:21
%S A308430 0,0,1,0,0,0,3,4,3,2,-1,1,3,3,1,1,-1,-3,0,1,4,3,4,5,8,9,8,7,6,7,2,6,
%T A308430 10,12,14,14,14,16,16,16,16,16,12,16,18,18,18,14,14,14,14,10,10,6,13,
%U A308430 16,19,20,23,26,27,30,31,30,31,30,31,34,33,32,35,34,31,30,27,22,25,26,29,30,31,32,29,30,27,24,27,28,27,24,23,18,15,12,9,4,-1,5,9,11
%N A308430 Number of 0's minus number of 1's among the edge truncated binary representations of the first n prime numbers.
%C A308430 By "edge truncated" we mean removing the first and last digit. For prime(3)=5 which has binary representation 101 edge truncating yields the string '0'. If there are 2 digits, then edge truncation yields the empty string ''. We count zero 1's and zero 0's in the empty string. The only cases of this are prime(1)=2 and prime(2)=3 which have binary representations 10 and 11.
%H A308430 Rémy Sigrist, <a href="/A308430/b308430.txt">Table of n, a(n) for n = 1..12251</a>
%H A308430 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a308/A308430.java">Java program</a> (github)
%H A308430 Jonas K. Sønsteby, <a href="/A308430/a308430_10.png">Graph of 200 terms.</a>
%H A308430 Jonas K. Sønsteby, <a href="/A308430/a308430_11.png">Graph of 1000 terms.</a>
%H A308430 Jonas K. Sønsteby, <a href="/A308430/a308430_12.png">Graph of 5000 terms.</a>
%H A308430 Jonas K. Sønsteby, <a href="/A308430/a308430_13.png">Graph of 10000 terms.</a>
%H A308430 Jonas K. Sønsteby, <a href="/A308430/a308430_14.png">Graph of 100000 terms.</a>
%F A308430 a(n) = a(n-1) + bitlength(prime(n)_2) - 2 * popcount(prime(n)_2) + 2, n > 1. - _Sean A. Irvine_, May 27 2019
%F A308430 a(n) = Sum_{k=2..n} (A035100(k) - 2*A014499(k) + 2) = Sum_{k=2..n} (A070939(A000040(k)) - 2*A000120(A000040(k)) + 2). - _Daniel Suteu_, Jul 13 2019
%o A308430 (Python)
%o A308430 import gmpy2
%o A308430 def dec2bin(x):
%o A308430     return str(bin(x))[2:]
%o A308430 def digitBalance(string):
%o A308430     s = 0
%o A308430     for char in string:
%o A308430         if int(char) > 0:
%o A308430             s -= 1
%o A308430         else:
%o A308430             s += 1
%o A308430     return s
%o A308430 N = 100 # number of terms
%o A308430 seq = [0]
%o A308430 prime = 2
%o A308430 for i in range(N-1):
%o A308430     prime = gmpy2.next_prime(prime)
%o A308430     binary = dec2bin(prime)
%o A308430     truncated = binary[1:-1]
%o A308430     term = seq[-1] + digitBalance(truncated)
%o A308430     seq.append(term)
%o A308430 print(seq) # _Jonas K. Sønsteby_, May 27 2019
%o A308430 (PARI) s=0; forprime (p=2, 541, print1 (s += #binary(p\2)+1-2*hammingweight(p\2) ", ")) \\ _Rémy Sigrist_, Jul 13 2019
%o A308430 (Sage)
%o A308430 def A308430list(b):
%o A308430     L = []; s = 0
%o A308430     for p in prime_range(2, b):
%o A308430         q = (p//2).digits(2)
%o A308430         s += 1 + len(q) - 2*sum(q)
%o A308430         L.append(s)
%o A308430     return L
%o A308430 print(A308430list(542)) # _Peter Luschny_, Jul 13 2019
%Y A308430 Cf. A004676, A095375, A014499, A177718, A296062.
%K A308430 sign,base,look
%O A308430 1,7
%A A308430 _Andrea Fornaciari_, May 26 2019