A035100 -id:A035100 - cp's OEIS frontend

A061716 Binary order of n-th prime.

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10

Offset: 1

Labos Elemer, Jun 20 2001

nonn

Apart from the first terms, the same as A035100. - R. J. Mathar, Oct 02 2008

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A029837, A000040, A003070, A045716, A045716, A036378.

Ceiling[Log2[Prime[Range[110]]]] (* Harvey P. Dale, Apr 12 2023 *)

a(n) = { logint(prime(n)-1, 2) + 1 } \\ Harry J. Smith, Jul 26 2009

a(n) = ceiling(log_2(prime(n))) = A029837(A000040(n)).

Offset changed from 0 to 1 by Harry J. Smith, Jul 26 2009

A084421 a(n) = A005187(A000040(n)).

Original entry on oeis.org

3, 4, 8, 11, 19, 23, 32, 35, 42, 54, 57, 71, 79, 82, 89, 102, 113, 117, 131, 138, 143, 153, 162, 174, 191, 198, 201, 209, 213, 222, 247, 259, 271, 274, 294, 297, 309, 322, 329, 341, 353, 357, 375, 383, 390, 393, 417, 439, 449, 453, 461, 471, 477, 495, 512

Offset: 1

Reinhard Zumkeller, Jun 26 2003

nonn

			A000040(6)=13 -> [13/2]=6 -> [6/2]=3 -> [3/2]=1
.......... 13 -> 13+6=19 -> 19+3=22 -> 22+1=23=a(6).

Cf. A000040, A005187, A035100.

A084421 := proc(n)
    A005187(ithprime(n)) ;
end proc:
seq(A084421(n),n=1..55) ; # R. J. Mathar, Jul 09 2016

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

Original entry on oeis.org

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

Andrea Fornaciari, May 26 2019

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.

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.

Cf. A004676, A095375, A014499, A177718, A296062.

s=0; forprime (p=2, 541, print1 (s += #binary(p\2)+1-2*hammingweight(p\2) ", ")) \\ Rémy Sigrist, Jul 13 2019

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

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

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

A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

Original entry on oeis.org

2, 257, 8519971, 36574494881, 140739702949921, 140773995710729, 140774004099109

Offset: 1

Gus Wiseman, Sep 27 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Conjecture: This sequence is infinite.

			The initial terms together with their binary indices:
                2: {2}
              257: {1,9}
          8519971: {1,2,6,9,18,24}
      36574494881: {1,6,8,16,18,27,32,36}
  140739702949921: {1,6,12,27,32,48}
  140773995710729: {1,4,9,12,18,32,36,48}
  140774004099109: {1,3,6,12,18,24,32,36,48}

Wikipedia, Geometric mean

A subset of A327368.
The binary weight of prime(n) is A014499(n), with binary length A035100(n).
Heinz numbers of partitions with integer mean: A316413.
Heinz numbers of partitions with integer geometric mean: A326623.
Heinz numbers with both: A326645.
Subsets with integer mean: A051293
Subsets with integer geometric mean: A326027
Subsets with both: A326643
Partitions with integer mean: A067538
Partitions with integer geometric mean: A067539
Partitions with both: A326641
Strict partitions with integer mean: A102627
Strict partitions with integer geometric mean: A326625
Strict partitions with both: A326029
Factorizations with integer mean: A326622
Factorizations with integer geometric mean: A326028
Factorizations with both: A326647
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Numbers whose binary indices have both: A327368
Cf. A000120, A029931, A034797, A048793, A070939, A096111, A291166, A326031, A326644, A326699/A326700.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Prime[Range[1000]],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

a(4)-a(7) from Giovanni Resta, Dec 01 2019

A227471 Position of first 0 in the binary representation of prime(n), starting the count of positions at 1 for the least significant bit.

Original entry on oeis.org

1, 3, 2, 4, 3, 2, 2, 3, 4, 2, 6, 2, 2, 3, 5, 2, 3, 2, 3, 4, 2, 5, 3, 2, 2, 2, 4, 3, 2, 2, 8, 3, 2, 3, 2, 4, 2, 3, 4, 2, 3, 2, 7, 2, 2, 4, 3, 6, 3, 2, 2, 5, 2, 3, 2, 4, 2, 5, 2, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 2, 2, 4, 5, 2, 3, 8, 2, 2, 2, 2, 3, 2, 5, 2, 4, 3, 2, 2, 2

Offset: 1

Juri-Stepan Gerasimov, Jul 12 2013

nonn

If no zero appears in the base-2 representation, the search "falls through" and addresses the (virtual) leading zero of that prime, see A035100. - R. J. Mathar, Jul 20 2013

Cf. A004676, A023512.

a(n) = 1+A023512(n). - Antti Karttunen, Jul 13 2013

cp's OEIS Frontend

A061716 Binary order of n-th prime.

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A084421 a(n) = A005187(A000040(n)).

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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.

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A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

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Mathematica

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A227471 Position of first 0 in the binary representation of prime(n), starting the count of positions at 1 for the least significant bit.

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