A014499 -id:A014499 - cp's OEIS frontend

A214179 Primes p such that least positive primitive root of p is equal to number of 1's in binary representation of p.

2, 3, 5, 7, 47, 103, 137, 157, 167, 239, 307, 397, 431, 433, 499, 521, 641, 647, 919, 997, 1163, 1303, 1429, 1543, 1753, 1811, 2063, 2081, 2087, 2111, 2377, 2591, 2711, 2833, 3119, 3181, 3217, 3719, 4079, 4153, 4211, 4271, 4273, 4297, 4549, 4643, 4721, 4931, 4993, 5399, 5857

Offset: 1

Juri-Stepan Gerasimov, Jul 07 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001918, A014499.

Select[Prime[Range[750]], PrimitiveRoot[#] == Plus@@IntegerDigits[#, 2] &] (* Alonso del Arte, Jul 09 2012 *)

A239695 Base 9 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 3, 5, 9, 3, 7, 5, 7, 5, 9, 11, 7, 13, 11, 13, 11, 15, 9, 15, 3, 9, 9, 5, 7, 11, 5, 9, 7, 11, 9, 11, 13, 15, 13, 3, 7, 5, 11, 5, 7, 9, 13, 7, 11, 15, 11, 13, 17, 15, 17, 11, 9, 7, 13, 7, 13, 9, 11, 13, 11, 15, 17, 13, 11, 9, 11, 13, 9, 15, 15, 13, 11

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-9 dominance order on the natural numbers.

			The fifth prime is 11, 11 in base 9 is (1,2) so a(5)=1+2=3.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053830, A239690, A239691, A239692, A239693, A239694.

[&+Intseq(NthPrime(n),9): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 9], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 9); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=9)) for i in primes_first_n(200)]

a(n) = A053830(A000040(n)).

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

A356876 Binary weight of the composite numbers (A002808).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 4, 1, 2, 2, 3, 3, 2, 3, 3, 4, 3, 4, 1, 2, 2, 3, 2, 3, 4, 2, 3, 3, 4, 4, 2, 3, 3, 4, 3, 4, 5, 3, 4, 4, 4, 5, 6, 1, 2, 2, 2, 3, 3, 2, 3, 4, 3, 4, 4, 2, 3, 3, 3, 4, 4, 5, 3, 4, 5, 4, 5, 5, 6, 2, 3, 4, 3, 4, 3, 4, 4, 4, 5, 6, 3, 4, 5, 4, 5, 5, 6, 4, 5, 5

Offset: 1

Karl-Heinz Hofmann, Oct 02 2022

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A002808, A000120, A014499.

DigitCount[#, 2, 1] & /@ Select[Range[125], CompositeQ] (* Amiram Eldar, Oct 03 2022 *)

forcomposite (k=0, 120, print1 (hammingweight(k),", ")) \\ Hugo Pfoertner, Oct 03 2022

from sympy import isprime
print([bin(k)[2:].count("1") for k in range(4, 123) if not isprime(k)])

a(n) = A000120(A002808(n)).

A360448 Indices of primes of the form p = 2^i + 2^j + 1, i > j > 0 (A081091).

Original entry on oeis.org

4, 5, 6, 8, 12, 13, 19, 21, 25, 32, 33, 44, 98, 106, 116, 136, 174, 191, 310, 313, 319, 565, 568, 1029, 1470, 1902, 2111, 3513, 3518, 3521, 4289, 6544, 12426, 13632, 15000, 23001, 23003, 23043, 23673, 43395, 43420, 43465, 45859, 62947, 82029, 82063, 91466, 155612, 155900, 295957, 564164

Offset: 1

M. F. Hasler, Mar 03 2023

nonn

Cf. A000720 (prime counting function), A081091 (primes of the form 2^i + 2^j + 1, i > j > 0), A014499 (Hamming weight of the n-th prime), A000040 (the primes), A000120 (Hamming weight).

Position[Prime[Range[600000]], ?(DigitCount[#, 2, 1] == 3 &)] // Flatten (* _Amiram Eldar, Mar 04 2023 *)
PrimePi@ Union@ Select[Flatten@ Table[2^i + 2^j + 1, {i, 0, 23}, {j, 0, i - 1}], PrimeQ] (* Michael De Vlieger, Mar 21 2023 *)

A360448(n) = primepi(A081091(n))
is_A360448(n) = hammingweight(prime(n))==3

from itertools import count, islice
from sympy import primepi, isprime
def A360448_gen(): # generator of terms
    for i in count(2):
        k = (1<A360448_list = list(islice(A360448_gen(),20)) # Chai Wah Wu, Mar 21 2023

a(n) = A000720(A081091(n)).

This sequence = { n | A000120(A000040(n)) = 3 }.

A095377 Values of k such that the total number of 1's in the binary expansions of the first k primes is a multiple of k.

Original entry on oeis.org

1, 4, 14, 43, 46, 141, 4900, 10264541, 10281244, 10281247, 10281248, 10281249, 10281266, 10281271, 10368575, 531439030, 1997778943, 412276655628

Offset: 1

Labos Elemer, Jun 07 2004

Integer solutions to {A095375(x)/x is an integer}.
a(18) > pi(10^12). - Donovan Johnson, May 03 2010
a(20) > 6.2*10^11. The first 19 ratios between the total number of 1's and k are 1, 2, 3, 4, 4, 5, 8, 14, 14, 14, 14, 14, 14, 14, 14, 17, 18, 22. - Giovanni Resta, May 08 2017

			n=14: the relevant list = {2,3,5...,41,43} = {10,11,101,...,11001,11011} contains 42 digits "1", and 42/14 = 3, so 14 is in the sequence.

Cf. A000120, A000788, A014499, A095375.

s=0; Reap[Do[s += DigitCount[Prime@n, 2][[1]]; If[Mod[s, n] == 0, Sow@ n], {n, 10^4}]][[2, 1]] (* Giovanni Resta, May 08 2017 *)

Integer solutions to {A095375(x)/x is an integer}.

a(16)-a(17) from Donovan Johnson, May 03 2010

a(18) from Giovanni Resta, May 08 2017

A169817 n-th prime with both a prime number of 0's and a prime number of 1's in binary representation minus n-th semiprime with both a prime number of 0's and a prime number of 1's in their binary representation.

Original entry on oeis.org

8, 9, 16, 19, 54, 77, 72, 71, 82, 72, 64, 66, 74, 79, 64, 63, 72, 77, 78, 93, 86, 88, 95, 102, 209, 218, 246, 245, 240, 258, 281, 278, 285, 304, 323, 238, 182, 187, 162, 142, 155, 136, 135, 124, 130, 139, 142, 138, 142, 134, 148, 166, 167, 174, 176, 168, 177, 174

Offset: 1

Juri-Stepan Gerasimov, May 25 2010

nonn

			a(1)=A144214(1)-A178350(1)=17-9=8.

Cf. A014499, A035103, A178064, A178065.

pn0Q[n_]:=PrimeQ[DigitCount[n,2,1]]&&PrimeQ[DigitCount[n,2,0]]; nn=600;Module[{ps=Select[Prime[Range[nn]],pn0Q],sps=Select[Range[nn], PrimeOmega[#]==2&&pn0Q[#]&],minlen},minlen=Min[Length[ps], Length[ sps]];First[#]-Last[#]&/@Thread[{Take[ps,minlen],Take[sps,minlen]}]] (* Harvey P. Dale, May 07 2012 *)

a(n)=A144214(n)-A178350(n).

Corrected (96 replaced by 86, all numbers from a(27) on replaced) by R. J. Mathar, Jun 04 2010

A362979 Square array, read by descending antidiagonals: row n lists the primes whose base-2 representation has exactly n ones, starting from n=3.

Original entry on oeis.org

7, 11, 23, 13, 29, 31, 19, 43, 47, 311

Offset: 3

Clark Kimberling, May 11 2023

			Corner:
  n=3:    7    11    13    19    37   41     67    73    97
  n=4:   23    29    43    53    71   83     89   101   113
  n=5:   31    47    59    61    79   103   107   109   151
  n=6:  311   317   347   349   359   373   461   467   571
The first four primes in row n=3 have these base-2 representations, respectively: 111, 1011, 1101, 10011.

Cf. A000040, A014499.

Cf. A019434 (row 2), A061712 (column 1), A081091 (row 3), A095077 (row 4).

t[n_] := Count[IntegerDigits[Prime[n], 2], 1]  (* A014499 *)
u = Table[t[n], {n, 1, 200}];
p[n_] := Flatten[Position[u, n]]
w = TableForm[Table[Prime[p[n]], {n, 3, 16}]]

New offset and edited by Michel Marcus, Jan 19 2024

cp's OEIS Frontend

A214179 Primes p such that least positive primitive root of p is equal to number of 1's in binary representation of p.

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A239695 Base 9 sum of digits of prime(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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A356876 Binary weight of the composite numbers (A002808).

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A360448 Indices of primes of the form p = 2^i + 2^j + 1, i > j > 0 (A081091).

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A095377 Values of k such that the total number of 1's in the binary expansions of the first k primes is a multiple of k.

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A169817 n-th prime with both a prime number of 0's and a prime number of 1's in binary representation minus n-th semiprime with both a prime number of 0's and a prime number of 1's in their binary representation.

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