A095062 -id:A095062 - cp's OEIS frontend

A095102 Odd primes p for which all sums Sum_{i=1..u} L(i/p) (with u ranging from 1 to (p-1)) are nonnegative, where L(i/p) is Legendre symbol of i and p, defined to be 1 if i is a quadratic residue (mod p) and -1 if i is a quadratic non-residue (mod p).

3, 7, 11, 23, 31, 47, 59, 71, 79, 83, 103, 131, 151, 167, 191, 199, 239, 251, 263, 271, 311, 359, 383, 419, 431, 439, 479, 503, 563, 599, 607, 647, 659, 719, 743, 751, 839, 863, 887, 911, 919, 971, 983, 991, 1031, 1039, 1063, 1091, 1103, 1151

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

All 4k+3 primes whose Legendre-vector (cf. A055094) forms a valid Dyck-path (cf. A014486).

T. D. Noe, Table of n, a(n) for n=1..1000
Peter Borwein, Stephen K.K. Choi and Michael Coons, Completely multiplicative functions taking values in {-1,1}, arXiv:0809.1691 [math.NT], 2008.
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A095100. Subset of A080114 (see comments there). Complement of A095103 in A002145.

Cf. A095092.

isMotzkin[n_, k_] := Module[{s = 0, r = True}, Do[s += JacobiSymbol[i, n]; If[s < 0, r = False; Break[]], {i, 1, k}]; r]; A095102[max_] := Select[ Range[3, max, 4], PrimeQ[#] && isMotzkin[#, Quotient[#, 2]]&]; A095102[1151] (* Jean-François Alcover, Feb 16 2018, after Peter Luschny *)

isok(m) = {if(!isprime(m-(m<3)), return(0)); my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; } \\ Jinyuan Wang, Jul 20 2020

def A095102_list(n) :
    def is_Motzkin(n, k):
        s = 0
        for i in (1..k):
            s += jacobi_symbol(i, n)
            if s < 0: return False
        return True
    P = filter(is_prime, range(3, n+1, 4))
    return filter(lambda m: is_Motzkin(m, m//2), P)
A095102_list(1151) # Peter Luschny, Aug 09 2012

a(n) = 4*A095272(n) + 3.

A095070 One-bit dominant primes, i.e., primes whose binary expansion contains more 1's than 0's.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 61, 71, 79, 83, 89, 101, 103, 107, 109, 113, 127, 151, 157, 167, 173, 179, 181, 191, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 311, 313, 317, 331, 347, 349, 359, 367, 373, 379, 383

Offset: 1

Antti Karttunen, Jun 01 2004

			23 is in the sequence because 23 is a prime and 23_10 = 10111_2. '10111' has four 1's and one 0. - _Indranil Ghosh_, Jan 31 2017

Indranil Ghosh, Table of n, a(n) for n = 1..20000
Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
MathOverflow, Primes with more ones than zeroes in their Binary expansion, 2012.

Intersection of A000040 and A072600.
Complement of A095075 in A000040.
Subsequence: A095073.
Cf. A095020.

Select[Prime[Range[70]], Plus@@IntegerDigits[#, 2] > Length[IntegerDigits[#, 2]]/2 &] (* Alonso del Arte, Jan 11 2011 *)
Select[Prime[Range[100]], Differences[DigitCount[#, 2]][[1]] < 0 &] (* Amiram Eldar, Jul 25 2023 *)

B(x) = {nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b1 > b0, return(1);, return(0););};
forprime(x = 3, 383, if(B(x), print1(x, ", "); ); ); \\ Washington Bomfim, Jan 11 2011

has(n)=hammingweight(n)>#binary(n)/2
select(has, primes(500)) \\ Charles R Greathouse IV, May 02 2013

# Program to generate the b-file
from sympy import isprime
i=1
j=1
while j<=200:
    if isprime(i) and bin(i)[2:].count("1")>bin(i)[2:].count("0"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Jan 31 2017

A095100 Integers m of the form 4k+3 for which all sums Sum_{i=1..u} J(i/m) (with u ranging from 1 to (m-1)) are nonnegative, where J(i/m) is Jacobi symbol of i and m.

Original entry on oeis.org

3, 7, 11, 15, 23, 27, 31, 35, 39, 47, 55, 59, 63, 71, 75, 79, 83, 87, 95, 103, 111, 119, 131, 135, 143, 151, 159, 167, 171, 175, 183, 191, 199, 215, 231, 239, 243, 251, 255, 263, 271, 279, 287, 295, 299, 303, 311, 319, 327, 335, 343, 351, 359, 363

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Integers whose Jacobi-vector forms a valid Motzkin-path.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Subset of A095102. Complement of A095101 in A004767.

Cf. A095090.

isMotzkin[n_, k_] := Module[{s = 0, r = True}, Do[s += JacobiSymbol[i, n]; If[s < 0, r = False; Break[]], {i, 1, k}]; r]; A095100[n_] := Select[4*Range[0, n+1]+3, isMotzkin[#, Quotient[#, 2]] &]; A095100[90] (* Jean-François Alcover, Oct 08 2013, translated from Sage *)

isok(m) = {if(m%4<3, return(0)); my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; } \\ Jinyuan Wang, Jul 20 2020

def is_Motzkin(n, k):
    s = 0
    for i in range(1, k + 1) :
        s += jacobi_symbol(i, n)
        if s < 0: return False
    return True
def A095100_list(n):
    return [m for m in range(3, n + 1, 4) if is_Motzkin(m, m // 2)]
A095100_list(363) # Peter Luschny, Aug 08 2012

a(n) = 4*A095274(n) + 3.

A095008 Number of 4k+3 primes (A002145) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 13, 22, 37, 71, 128, 231, 440, 807, 1519, 2872, 5371, 10204, 19341, 36759, 70179, 134241, 256856, 492936, 947272, 1822615, 3513691, 6781495, 13103816, 25348667, 49092241, 95168205, 184661253, 358636497, 697094872, 1356052491, 2639893495, 5142817901

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A002145, A036378, A095007, A095010, A095012, A095092, A095093.

a(n) = A036378(n) - A095007(n) = A095010(n) + A095012(n) = A095092(n) + A095093(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A095075 Primes in whose binary expansion the number of 1-bits is less than or equal to number of 0-bits.

Original entry on oeis.org

2, 17, 37, 41, 67, 73, 97, 131, 137, 139, 149, 163, 193, 197, 257, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 521, 523, 541, 547, 557, 563, 569, 577, 587, 593, 601, 613, 617, 641, 643, 647, 653, 659, 661, 673, 677, 709, 769, 773, 787

Offset: 1

Antti Karttunen, Jun 01 2004

			From _Indranil Ghosh_, Feb 03 2017: (Start)
17 is in the sequence because 17_10 = 10001_2. '10001' has two 1's and three 0's.
37 is in the sequence because 37_10 = 100101_2. '100101' has three 1's and 3 0's. (End)

Indranil Ghosh, Table of n, a(n) for n = 1..25000
Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.

Complement of A095070 in A000040.

Cf. A095055.

Select[Prime[Range[150]], Differences[DigitCount[#, 2]][[1]] >= 0 &] (* Amiram Eldar, Jul 25 2023 *)
Select[Prime[Range[150]],DigitCount[#,2,1]<=DigitCount[#,2,0]&] (* Harvey P. Dale, Sep 27 2023 *)

B(x) = {nB = floor(log(x)/log(2)); z1 = 0; z0 = 0;
for(i = 0, nB, if(bittest(x,i), z1++;, z0++;); );
if(z1 <= z0, return(1);, return(0););};
forprime(x = 2, 787, if(B(x), print1(x, ", "); ); );
\\ Washington Bomfim, Jan 11 2011

from sympy import isprime
i=1
j=1
while j<=250:
    if isprime(i) and bin(i)[2:].count("1")<=bin(i)[2:].count("0"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

A095071 Zero-bit dominant primes, i.e., primes whose binary expansion contains more 0's than 1's.

Original entry on oeis.org

17, 67, 73, 97, 131, 137, 193, 257, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 521, 523, 547, 577, 593, 641, 643, 673, 769, 773, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1091, 1093, 1097, 1109, 1123, 1129, 1153, 1163, 1171

Offset: 1

Antti Karttunen, Jun 01 2004

			73 is in the sequence because 73 is a prime and 73_10 = 1001001_2. '1001001' has four 0's and one 1. - _Indranil Ghosh_, Jan 31 2017

Indranil Ghosh, Table of n, a(n) for n = 1..20000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Complement of A095074 in A000040. Subset: A095072. Cf. A095019.

Reap[Do[p=Prime[k];id=IntegerDigits[p,2];n=Length@id;If[Count[id,0]>n/2,Sow[p]],{k,200}]][[2,1]]
(* Zak Seidov *)
Select[Prime[Range[200]],DigitCount[#,2,0]>DigitCount[#,2,1]&] (* Harvey P. Dale, Nov 28 2024 *)

B(x) = { nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b0 > b1, return(1);, return(0););};
forprime(x = 2, 1171, if(B(x), print1(x, ", "); ); ); \\ Washington Bomfim, Jan 11 2011

{forprime(p=2,1171,nB=floor(log(p)/log(2));
sum(i=0,nB,bittest(p,i))<=nB/2&print1(p,","))} \\ Zak Seidov, Jan 11 2011

#Program to generate the b-file
from sympy import isprime
i=1
j=1
while j<=200:
    if isprime(i) and bin(i)[2:].count("0")>bin(i)[2:].count("1"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Jan 31 2017

A095072 Primes in whose binary expansion the number of 0-bits is one more than the number of 1-bits.

Original entry on oeis.org

17, 67, 73, 97, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 1039, 1051, 1063, 1069, 1109, 1123, 1129, 1163, 1171, 1187, 1193, 1201, 1249, 1291, 1301, 1321, 1361, 1543, 1549, 1571, 1609, 1667, 1669, 1697, 1801, 4127, 4157, 4211, 4217

Offset: 1

Antti Karttunen, Jun 01 2004

A010051(a(n)) = 1 and A037861(a(n)) = 1. - Reinhard Zumkeller, Mar 31 2015

			97 is in the sequence because 97 is a prime and 97_10 = 1100001_2. The number of 0's in 1100001 is 4 and the number of 1's is 3. - _Indranil Ghosh_, Jan 31 2017

Indranil Ghosh, Table of n, a(n) for n = 1..20000 (terms 1..1000 from Reinhard Zumkeller)
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A031444. Subset of A095071.
Cf. A095052.
Cf. A010051, A037861.

a095072 n = a095072_list !! (n-1)
a095072_list = filter ((== 1) . a010051' . fromIntegral) a031444_list
-- Reinhard Zumkeller, Mar 31 2015

Select[Prime[Range[500]], Differences[DigitCount[#, 2]] == {1} &]

isA095072(n)=my(v=binary(n));#v==2*sum(i=1,#v,v[i])+1&&isprime(n)

forprime(p=2, 4250, v=binary(p); s=0; for(k=1, #v, s+=if(v[k]==0,+1,-1)); if(s==1,print1(p,", ")))

#Program to generate the b-file
from sympy import isprime
i=1
j=1
while j<=200:
    if isprime(i) and bin(i)[2:].count("0")-bin(i)[2:].count("1")==1:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Jan 31 2017

A095298 Sum of 1-bits between the most and least significant bits summed for all primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 2, 8, 15, 30, 67, 154, 302, 611, 1280, 2546, 5207, 10447, 21123, 42783, 85726, 173102, 347243, 698544, 1401784, 2813930, 5644165, 11328192, 22712057, 45538473, 91288241, 182965151, 366691833, 734702678, 1471976078, 2948741819

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratio a(n)/A036378(n) (i.e. average number of 1-bits in range ]most significant bit,least significant bit[ of primes p which 2^n < p < 2^(n+1)) grows as: 0, 0.5, 1, 1.6, 2.142857, 2.307692, 2.913043, 3.581395, 4.026667, 4.459854, 5.019608, 5.487069, 5.97133, 6.480769, 6.971287, 7.493957, 7.975254, 8.489554, 8.987783, 9.492893, 9.98877, 10.491283, 10.987107, 11.49116, 11.990823, 12.490859, 12.990533, 13.491108, 13.991985, 14.491881, 14.992221, 15.492331, 15.992713.

Ratio of that average compared to (n-1)/2 (the expected value of that same sum computed for all odd numbers in the same range) converges as: 1, 1, 1, 1.066667, 1.071429, 0.923077, 0.971014, 1.023256, 1.006667, 0.991079, 1.003922, 0.997649, 0.995222, 0.997041, 0.995898, 0.999194, 0.996907, 0.998771, 0.998643, 0.999252, 0.998877, 0.99917, 0.998828, 0.999231, 0.999235, 0.999269, 0.999272, 0.999341, 0.999427, 0.99944, 0.999481, 0.999505, 0.999545.

			a(1)=0, as only prime in range ]2,4] is 3, 11 in binary which has no space between its most and least significant bits. a(2)=1, as in that range there are two primes 5 (101 in binary) and 7 (111 in binary) and summing their middle bits we get 1. a(3)=2, as there are again two primes, 11 (1011 in binary) and 13 (1101 in binary) and summing the bits in the middle we get total 2.

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

A095297, A095334. Cf. also A095353 (similar sums and ratios computed in Fibonacci number system).

A095005 Number of odious primes (A027697) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 2, 2, 5, 8, 19, 20, 48, 75, 160, 242, 505, 835, 1761, 2799, 5890, 10250, 20921, 36872, 74316, 134816, 267749, 492286, 977207, 1823657, 3598657, 6779899, 13336543, 25358424, 49763462, 95140695, 186504600, 358630024, 702300885, 1356118149, 2654709953, 5142968571

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A027697, A036378, A095006.

Table[Count[Prime@ Range[PrimePi[2^n] + 1, PrimePi[2^(n + 1) - 1]], k_ /; OddQ@ First@ DigitCount[k, 2]], {n, 24}] (* Michael De Vlieger, Feb 25 2017 *)

a(n) = #select(x->((hammingweight(x)%2)==1),primes([2^n+1,2^(n+1)])); \\ Michel Marcus, Feb 26 2017

a(n) = A036378(n) - A095006(n).

a(34)-a(38) from Amiram Eldar, Jun 20 2024

cp's OEIS Frontend

A095018 a(n) is the number of primes p which have exactly n zeros and n ones when written in binary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A095102 Odd primes p for which all sums Sum_{i=1..u} L(i/p) (with u ranging from 1 to (p-1)) are nonnegative, where L(i/p) is Legendre symbol of i and p, defined to be 1 if i is a quadratic residue (mod p) and -1 if i is a quadratic non-residue (mod p).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A095070 One-bit dominant primes, i.e., primes whose binary expansion contains more 1's than 0's.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

A095100 Integers m of the form 4k+3 for which all sums Sum_{i=1..u} J(i/m) (with u ranging from 1 to (m-1)) are nonnegative, where J(i/m) is Jacobi symbol of i and m.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A095008 Number of 4k+3 primes (A002145) in range ]2^n,2^(n+1)].

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A095075 Primes in whose binary expansion the number of 1-bits is less than or equal to number of 0-bits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A095071 Zero-bit dominant primes, i.e., primes whose binary expansion contains more 0's than 1's.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python