A308430 Number of 0's minus number of 1's among the edge truncated binary representations of the first n prime numbers.
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, 10, 12, 14, 14, 14, 16, 16, 16, 16, 16, 12, 16, 18, 18, 18, 14, 14, 14, 14, 10, 10, 6, 13, 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
Offset: 1
Links
- Rémy Sigrist, Table of n, a(n) for n = 1..12251
- Sean A. Irvine, Java program (github)
- Jonas K. Sønsteby, Graph of 200 terms.
- Jonas K. Sønsteby, Graph of 1000 terms.
- Jonas K. Sønsteby, Graph of 5000 terms.
- Jonas K. Sønsteby, Graph of 10000 terms.
- Jonas K. Sønsteby, Graph of 100000 terms.
Programs
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PARI
s=0; forprime (p=2, 541, print1 (s += #binary(p\2)+1-2*hammingweight(p\2) ", ")) \\ Rémy Sigrist, Jul 13 2019
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Python
import gmpy2 def dec2bin(x): return str(bin(x))[2:] def digitBalance(string): s = 0 for char in string: if int(char) > 0: s -= 1 else: s += 1 return s N = 100 # number of terms seq = [0] prime = 2 for i in range(N-1): prime = gmpy2.next_prime(prime) binary = dec2bin(prime) truncated = binary[1:-1] term = seq[-1] + digitBalance(truncated) seq.append(term) print(seq) # Jonas K. Sønsteby, May 27 2019 -
Sage
def A308430list(b): L = []; s = 0 for p in prime_range(2, b): q = (p//2).digits(2) s += 1 + len(q) - 2*sum(q) L.append(s) return L print(A308430list(542)) # Peter Luschny, Jul 13 2019
Formula
a(n) = a(n-1) + bitlength(prime(n)2) - 2 * popcount(prime(n)_2) + 2, n > 1. - _Sean A. Irvine, May 27 2019
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