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 A308430 Number of 0's minus number of 1's among the edge truncated binary representations of the first n prime numbers. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS 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. 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. FORMULA a(n) = a(n-1) + bitlength(prime(n)_2) - 2 * popcount(prime(n)_2) + 2, n > 1. - Sean A. Irvine, May 27 2019 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 PROG (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 =  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 (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 print(A308430list(542)) # Peter Luschny, Jul 13 2019 CROSSREFS Cf. A004676, A095375, A014499, A177718, A296062. Sequence in context: A201935 A225445 A167877 * A280136 A258451 A332412 Adjacent sequences:  A308427 A308428 A308429 * A308431 A308432 A308433 KEYWORD sign,base,look AUTHOR Andrea Fornaciari, May 26 2019 STATUS approved

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Last modified July 15 23:32 EDT 2020. Contains 335774 sequences. (Running on oeis4.)