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A308430
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Number of 0's minus number of 1's among the edge truncated binary representations of the first n prime numbers.
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1
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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
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OFFSET
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1,7
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COMMENTS
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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.
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LINKS
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FORMULA
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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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PROG
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(Python 3)
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)
(PARI) s=0; forprime (p=2, 541, print1 (s += #binary(p\2)+1-2*hammingweight(p\2) ", ")) \\ Rémy Sigrist, Jul 13 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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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