A130911 -id:A130911 - cp's OEIS frontend

A027699 Evil primes: primes with even number of 1's in their binary expansion.

3, 5, 17, 23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 257, 263, 269, 277, 281, 293, 311, 317, 337, 347, 349, 353, 359, 373, 383, 389, 401, 449, 461, 467, 479, 503, 509, 523, 547, 571, 593, 599, 619, 643, 673, 683, 691, 739, 751, 773, 797, 811

Offset: 1

N. J. A. Sloane

Comment from Vladimir Shevelev, Jun 01 2007: Conjecture: If pi_1(m) is the number of a(n) not exceeding m and pi_2(m) is the number of A027697(n) not exceeding m then pi_1(m) <= smaller than pi_2(m) for all natural m except m=5 and m=6. I verified this conjecture up to 10^9. Moreover I conjecture that pi_2(m)-pi_1(m) tends to infinity with records at the primes m=2, 13, 41, 61, 67, 79, 109, 131, 137, ...

T. D. Noe, Table of n, a(n) for n = 1..10000
E. Fouvry, C. Mauduit, Sommes des chiffres et nombres presque premiers, (French) [Sums of digits and almost primes] Math. Ann. 305 (1996), no. 3, 571--599. MR1397437 (97k:11029).
V. Shevelev, A conjecture on primes and a step towards justification, arXiv:0706.0786 [math.NT], 2007.

Cf. A027697, A066148, A066149.
Cf. A001969 (evil numbers), A129771 (evil odd numbers)
Cf. A130911 (prime race between evil primes and odious primes).

Select[Prime[Range[200]], EvenQ[Count[IntegerDigits[ #,2],1]]&] (* T. D. Noe, Jun 12 2007 *)

forprime(p=1,999,norml2(binary(p))%2 || print1(p","))

isA027699(p)=isprime(p) && !bittest(norml2(binary(p)),0) \\ M. F. Hasler, Dec 12 2010

from sympy import isprime
def ok(n): return bin(n).count("1")%2 == 0 and isprime(n)
print([k for k in range(812) if ok(k)]) # Michael S. Branicky, Jun 27 2022

More terms from Erich Friedman

A156549 Race between primes having an odd/even number of zeros in their binary representation.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 7, 8, 9, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 21, 22, 21, 22, 21, 22, 21, 22, 23, 24, 25, 26, 25, 26, 25, 26, 27, 26, 27, 26, 25, 24, 23, 22

Offset: 1

T. D. Noe, Feb 09 2009

See A066148 and A066149 for primes with an even/odd number of zeros in their binary representation. Sequence A130911 shows the race between primes having an odd/even number of ones in their binary representation. In this sequence (and A130911), it appears that the primes with an odd number of zeros (or ones) dominate the primes with an even number of zeros (or ones). In general, it appears that the sequences grow for primes having an odd number of bits and "rest" for primes having an even number of bits.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A066148, A066149, A130911.

cnt=0; Table[p=Prime[n]; If[OddQ[Count[IntegerDigits[p,2],0]], cnt++, cnt-- ]; cnt, {n,100}]
Accumulate[Table[If[OddQ[DigitCount[p,2,0]],1,-1],{p,Prime[Range[90]]}]] (* Harvey P. Dale, Apr 02 2025 *)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==0, s++));return(s%2)}; nO=0;nE=0; forprime(p=2,435,if(f(p), nO++, nE++); an = nO-nE; print1(an,", ")) \\ Washington Bomfim, Jan 14 2011

a(n) = (number of primes having an odd number of zeros <= prime(n)) - (number of primes having an even number of zeros <= prime(n))

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

Original entry on oeis.org

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

A199339 a(n) = number of primes with an even digit sum among the first n primes minus the number with an odd digit sum.

Original entry on oeis.org

1, 0, -1, -2, -1, 0, 1, 2, 1, 0, 1, 2, 1, 0, -1, 0, 1, 0, -1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 5, 4, 5, 4, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0, -1, -2, -1, -2, -3, -4, -3, -2, -3, -2, -1, -2, -3, -2, -1, -2, -3, -2, -1, -2, -3, -4, -5, -6, -5, -4, -5, -6, -5, -6

Offset: 1

M. F. Hasler, Nov 14 2011

			a(1)=1 because the first prime has an even sum of digits.
a(2)=0, a(3)=-1, a(4)=-2 because the following primes (3,5,7) have odd sum of digits.
a(5)=-1, a(6)=0, a(7)=1, a(8)=2 because the 5th, 6th, 7th and 8th prime (11, 13, 17, 19) have an even sum of digits.

T. D. Noe, Table of n, a(n) for n = 1..10000
CNRS Press release, The sum of digits of prime numbers is evenly distributed, May 12, 2010.
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 12, 2010.

Cf. A007605, A007953, A130911, A200259, A200260, A119449, A119450, A200261, A200262, A200263, A200264.

a[1] := 1; a[n_] := a[n] = a[n - 1] + (-1)^(Plus@@IntegerDigits[Prime[n]]); Table[a[n], {n, 74}] (* Alonso del Arte, Nov 14 2011 *)

s=0;vector(90,n,s+=(-1)^A007953(prime(n)))

a(n)=sum_{k=1...n} (-1)^A007605(n).

Equals A200262 - A200264.

A361071 Let c1(p) be the number of primes <= p with an odd number of 1's in base 2, and let c2(p) be the number of primes <= p with an even number of 1's in base 2. a(n) is the least prime p such that abs(c1(p) - c2(p)) >= n.

Original entry on oeis.org

2, 13, 41, 61, 67, 79, 109, 131, 137, 173, 179, 181, 191, 193, 211, 223, 227, 229, 233, 239, 241, 251, 587, 613, 617, 641, 653, 659, 661, 719, 727, 733, 761, 769, 829, 953, 967, 971, 1009, 1021, 1039, 1069, 1087, 1193, 1201, 1213, 1697, 1721, 1753, 1759, 1777, 1783, 1787

Offset: 1

Jean-Marc Rebert, Mar 01 2023

			a(1) = 2, because c1(2) = 1 and c2(2) = 0, so abs(c1(2) - c2(2)) = 1 >= 1, and no lesser prime satisfies this.

Cf. A130911, A027697, A027699.

{ r = 0; n = 1; forprime (p = 2, 1787, r += (-1)^hammingweight(p); if (n==abs(r), print1 (p", "); n++;);); } \\ Rémy Sigrist, Mar 01 2023

cp's OEIS Frontend

A027699 Evil primes: primes with even number of 1's in their binary expansion.

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A156549 Race between primes having an odd/even number of zeros in their binary representation.

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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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A199339 a(n) = number of primes with an even digit sum among the first n primes minus the number with an odd digit sum.

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A361071 Let c1(p) be the number of primes <= p with an odd number of 1's in base 2, and let c2(p) be the number of primes <= p with an even number of 1's in base 2. a(n) is the least prime p such that abs(c1(p) - c2(p)) >= n.

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