A268477 -id:A268477 - 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

A268415 Balanced odious numbers: numbers with an odd number of runs of 1's in their binary expansion.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 21, 24, 28, 30, 31, 32, 37, 41, 42, 43, 45, 48, 53, 56, 60, 62, 63, 64, 69, 73, 74, 75, 77, 81, 82, 83, 84, 86, 87, 89, 90, 91, 93, 96, 101, 105, 106, 107, 109, 112, 117, 120, 124, 126, 127, 128, 133, 137, 138, 139, 141

Offset: 1

Vladimir Shevelev, Feb 04 2016

In the balanced binary system the sequence A268411 plays the role of the Thue-Morse sequence (A010060). Therefore we call the balanced odious numbers those numbers n for which A268411(n) = 1.

Complement of A268412.

			77 is a member because its binary expansion (1001101) has 3 runs of 1's, and 3 is odd.

Peter J. C. Moses (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..8129
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Positions of odd terms in A069010.
Cf. A010060, A000069, A268411.
Cf. A268412 (complement).
Cf. A268382 (the least monotonic left inverse).
Cf. A268477 (primes in this sequence).

balancedBinary:=Join[#,{0}]-Join[{0},#]&[IntegerDigits[#,2]]&;
Flatten[Position[Map[Mod[Count[balancedBinary[#],1],2]&,Range[0,100]],1]-1] (* Peter J. C. Moses, Feb 04 2016 *)
Select[Range[200],OddQ[Count[Split[IntegerDigits[#,2]],?(MemberQ[ #,1]&)]]&] (* _Harvey P. Dale, Mar 31 2019 *)

A268415_list = [i for i in range(10**6) if len(list(filter(bool,format(i,'b').split('0')))) % 2] # Chai Wah Wu, Mar 01 2016

(define A268415 (ZERO-POS 1 1 (COMPOSE -1+ A268411))) ;; requires also my IntSeq-library. - Antti Karttunen, Feb 05 2016

Other identities. For all n >= 1:

A268382(a(n)) = n.

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

A268476 Balanced evil primes: primes with an even number of runs of 1's in their binary expansion.

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 29, 47, 59, 61, 67, 71, 79, 97, 103, 113, 131, 149, 173, 181, 191, 193, 199, 223, 227, 239, 241, 251, 257, 263, 271, 277, 293, 331, 337, 347, 349, 373, 383, 421, 449, 463, 479, 487, 499, 503, 509, 557, 587, 593, 599, 601, 613, 617, 619

Offset: 1

Vladimir Shevelev, Feb 05 2016

Primes in A268412. Complement of A268477.

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. A027699, A268412, A268477.

Select[Prime@ Range@ 120, EvenQ@ Length[Split@ IntegerDigits[#, 2] /. {0, _} -> Nothing] &] (* Michael De Vlieger, Feb 08 2016 *)

from sympy import prime
A268476_list = [p for p in (prime(i) for i in range(1,10**6)) if not 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

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

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A268415 Balanced odious numbers: numbers with an odd number of runs of 1's in their binary expansion.

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A268476 Balanced evil primes: primes with an even number of runs of 1's in their binary expansion.

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Mathematica

Python

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