A268476 -id:A268476 - cp's OEIS frontend

A268483 Primes p such that the numbers of primes not exceeding p in A268476 and A268477 are equal.

13, 43, 53, 139, 151, 193, 199, 223, 229, 239, 317, 397, 4751, 4889, 4909, 4937, 4951, 4967, 5011, 5023, 5077, 5087, 5113, 5297, 5351, 5419, 6007, 6053, 6211, 6247, 6301, 6317, 6343, 6857, 9209, 9421, 9473, 9491, 10937, 11047, 11329, 11399, 11423, 11443, 11491

Offset: 1

Vladimir Shevelev, Feb 05 2016

In contrast to the analogous sequence for odious and evil primes (A027697, A027699), which, as we conjecture, consists of only primes 3,7,29 (see also our 2007-conjecture in A027697, A027699), here we conjecture that the sequence is infinite.

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Cf. A027697, A027699, A130911, A268476, A268477.

lim = 1500; s = Select[Prime@ Range@ lim, EvenQ@ Length[Split@ IntegerDigits[#, 2] /. {0, _} -> Nothing] &]; t = Select[Prime@ Range@ lim, OddQ@ Length[Split@ IntegerDigits[#, 2] /. {0, _} -> Nothing] &] ; Select[Prime@ Range@ lim, Count[s, p_ /; p <= #] == Count[t, q_ /; q <= #] &] (* Michael De Vlieger, Feb 08 2016 *)

More terms from Peter J. C. Moses, Feb 05 2016

A268412 Balanced evil numbers: numbers with an even number of runs of 1's in their binary expansion.

Original entry on oeis.org

0, 5, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 38, 39, 40, 44, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 70, 71, 72, 76, 78, 79, 80, 85, 88, 92, 94, 95, 97, 98, 99, 100, 102, 103, 104, 108, 110, 111, 113, 114

Offset: 0

Vladimir Shevelev, Feb 04 2016

In balanced binary system the sequence A268411 plays role of Thue-Morse sequence (A010060). Therefore, we call the balanced evil numbers those numbers n for which A268411(n) = 0.

			In binary representation 19=10011 has an even number (two) of runs of 1's. So, 19 is a member.

Peter J. C. Moses (terms 0..999) & Antti Karttunen, Table of n, a(n) for n = 0..8256
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Positions of even terms in A069010.
Cf. A010060, A001969, A268411.
Cf. A268415 (complement).
Cf. A268383 (the least monotonic left inverse).
Cf. A268476 (primes in this sequence).

balancedBinary:=Join[#,{0}]-Join[{0},#]&[IntegerDigits[#,2]]&;
Flatten[Position[Map[Mod[Count[balancedBinary[#],1],2]&,Range[0,100]],0,1]-1] (* Peter J. C. Moses, Feb 04 2016 *)

A268412_list = [i for i in range(10**6) if not len(list(filter(bool,format(i,'b').split('0')))) % 2] # Chai Wah Wu, Mar 01 2016

Other identities. For all n >= 0:

A268383(a(n)) = n.

More terms from Peter J. C. Moses, Feb 04 2016

A268477 Balanced odious primes: primes with an odd number of runs of 1's in their binary expansion.

Original entry on oeis.org

2, 3, 7, 31, 37, 41, 43, 53, 73, 83, 89, 101, 107, 109, 127, 137, 139, 151, 157, 163, 167, 179, 197, 211, 229, 233, 269, 281, 283, 307, 311, 313, 317, 353, 359, 367, 379, 389, 397, 401, 409, 419, 431, 433, 439, 443, 457, 461, 467, 491, 521, 523, 541, 547, 563

Offset: 1

Vladimir Shevelev, Feb 05 2016

Primes from A268415.
According to our 2007-conjecture, if pi_1(m) is the number of evil primes (A027699) not exceeding m and pi_2(m) is the number of odious primes (A027697) not exceeding m, then pi_1(m)<=pi_2(m) for all natural m except m=5 and m=6.
In the "balance" case of A268476,A268477, most likely, none of two types of primes
is in the majority beginning with any place.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Cf. A027697, A268415, A268476.

Select[Prime@ Range@ 108, OddQ@ Length[Split@ IntegerDigits[#, 2] /. {0, _} -> Nothing] &] (* Michael De Vlieger, Feb 08 2016 *)

from sympy import prime
A268477_list = [p for p in (prime(i) for i in range(1,10**6)) if len(list(filter(bool,format(p,'b').split('0')))) % 2] # Chai Wah Wu, Mar 01 2016

More terms from Peter J. C. Moses, Feb 05 2016

cp's OEIS Frontend

A268483 Primes p such that the numbers of primes not exceeding p in A268476 and A268477 are equal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A268412 Balanced evil numbers: numbers with an even number of runs of 1's in their binary expansion.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A268477 Balanced odious primes: primes with an odd number of runs of 1's in their binary expansion.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions