A027697 -id:A027697 - cp's OEIS frontend

A230120 a(n) is the number of evil integers (A001969) not exceeding n and prime to n.

0, 0, 0, 1, 1, 1, 3, 2, 1, 2, 5, 1, 6, 3, 0, 4, 7, 2, 9, 3, 4, 5, 10, 3, 8, 6, 5, 7, 13, 3, 15, 8, 6, 8, 12, 4, 18, 9, 7, 8, 20, 4, 20, 10, 5, 11, 23, 5, 24, 9, 8, 12, 25, 6, 19, 14, 11, 14, 29, 5, 30, 15, 12, 16, 22, 7, 33, 15, 12, 12, 34, 8, 36, 18, 10, 18

Offset: 1

Vladimir Shevelev, Oct 10 2013

See comment in A230070, taking into account the equality a(n) = phi(n) - A230070(n), where phi(n) is Euler totient function (A000010).

Georg Fischer, Table of n, a(n) for n = 1..5000

Cf. A000010, A001969, A230070, A027697, A027699.

upTo[n_] := Block[{c, i, e = Select[Range[n], EvenQ@ DigitCount[#, 2, 1] &]}, Table[c = 0; i = 1; While[i <= Length@ e && e[[i]] < k, c += Boole@ CoprimeQ[e[[i]], k]; i++]; c, {k, n}]]; upTo[100] (* Giovanni Resta, Apr 14 2025 *)

a(n) = sum(k = 1, n, gcd(k, n) == 1 && !(hammingweight(k) % 2)); \\ Amiram Eldar, Nov 10 2024

For odd evil prime p (A027699), a(p) = (p-3)/2; for odd odious prime p (A027697), a(p) = (p-1)/2.

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

A127977 The minimum excess in the prime race of odious primes versus evil primes in the interval (2^(n-1),2^n).

Original entry on oeis.org

0, 1, 4, 7, 13, 19, 39, 53, 104, 138, 251, 334, 590, 715, 1353, 1855, 3659, 5221, 10484, 14933, 27491, 35474, 68816, 97342, 186405, 265255

Offset: 5

Jonathan Vos Post, Jun 07 2007

Shevelev conjectures (p.2) that for all natural numbers n other than 5 and 6, the number of evil primes not exceeding n <= the number of odious primes not exceeding n. Odious primes are A027697. Evil primes are A027699.

			OdiPrimePi(x) for x >= 32 is 6, 6, 6, 6, 6, 7, 7, 7, 7, 8,.. and EviPrimePi(x) for x>=32 is 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6,...
The difference OdiPrimePi(x)-EviPrimePi(x) is 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3,.. so the minimum of the difference in the interval 2^(6-1)..2^6 is 1, yielding a(6)=1.

Vladimir Shevelev, A Conjecture on Primes and a Step towards Justification, arXiv:0706.0786 [math.NT], 2007. See table 1, p. 2.
Vladimir Shevelev, On excess of the odious primes, arXiv:0707.1761 [math.NT], 2007.

Cf. A000069, A001969, A027697, A027699, A095005, A095006.

read("transforms") ; # see oeis.org/transforms.txt
isA000069 := proc(n) type(wt(n),'odd') ; end proc;
isA027697 := proc(n) isprime(n) and isA000069(n) ; end proc:
isA027699 := proc(n) isprime(n) and not isA000069(n) ; end proc:
odiPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027697(n) then an1+1 ; else an1 ; end if; end if; end proc:
eviPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027699(n) then an1+1 ; else an1 ; end if; end if; end proc:
oddPi := proc(n) odiPi(n)-eviPi(n) ; end proc:
A127977 := proc(n) local a,x ; a := 2^(n+1) ; for x from 2^(n-1)+1 to 2^n-1 do a := min(a,oddPi(x)) ; end do: a; end proc:
for n from 5 do print(n,A127977(n)) ; end do; # R. J. Mathar, Sep 03 2011

wt[n_] := DigitCount[n, 2, 1];
isA000069[n_] := OddQ[wt[n]];
isA027697[n_] := PrimeQ[n] && isA000069[n];
isA027699[n_] := PrimeQ[n] && !isA000069[n];
odiPi[n_] := odiPi[n] = If[n==0, 0, an1 = odiPi[n-1]; If[isA027697[n], an1+1, an1]];
eviPi[n_] := eviPi[n] = If[n==0, 0, an1 = eviPi[n-1]; If[isA027699[n], an1+1, an1]];
oddPi[n_] := odiPi[n] - eviPi[n];
A127977[n_] := Module[{a, x}, a = 2^(n+1); For[x = 2^(n-1)+1, x <= 2^n-1, x++, a = Min[a, oddPi[x]]]; a];
Table[an = A127977[n]; Print[an]; an, {n, 5, 30}] (* Jean-François Alcover, Jan 23 2018, after R. J. Mathar *)

Published numbers corrected and checked up to n=23 by R. J. Mathar, Sep 03 2011

A173209 Partial sums of A000069.

Original entry on oeis.org

1, 3, 7, 14, 22, 33, 46, 60, 76, 95, 116, 138, 163, 189, 217, 248, 280, 315, 352, 390, 431, 473, 517, 564, 613, 663, 715, 770, 826, 885, 946, 1008, 1072, 1139, 1208, 1278, 1351, 1425, 1501, 1580, 1661, 1743, 1827, 1914, 2002, 2093, 2186, 2280, 2377, 2475, 2575

Offset: 1

Jonathan Vos Post, Feb 12 2010

Partial sums of odious numbers. Second differences give A007413. The subsequence of prime partial sums of odious numbers begins: 3, 7, 163, 431, 613, 2377, 3691, which is a subsequence of A027697. The subsequence of odious partial sums of odious numbers begins: 1, 7, 14, 22, 76, 138, 217, 280, 352, 431, 517, 613, 770, 885.

			a(65) = 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 19 + 21 + 22 + 25 + 26 + 28 + 31 + 32 + 35 + 37 + 38 + 41 + 42 + 44 + 47 + 49 + 50 + 52 + 55 + 56 + 59 + 61 + 62 + 64 + 67 + 69 + 70 + 73 + 74 + 76 + 79 + 81 + 82 + 84 + 87 + 88 + 91 + 93 + 94 + 97 + 98 + 100 + 103 + 104 + 107 + 109 + 110 + 112 + 115 + 117 + 118 + 121 + 122 + 124 + 127 + 128.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.

Cf. A000069, A000079, A000120, A000773, A000788, A001969, A002042, A002089, A007413, A019568, A023416, A027697, A059015, A083420, A010060, A132679, A133009.

Accumulate@ Select[Range[100], OddQ@ DigitCount[#, 2, 1] &] (* Michael De Vlieger, Nov 01 2022 *)

a(n)=n^2-(n+1)\2+(n%2&&hammingweight(n)%2) \\ Charles R Greathouse IV, Mar 22 2013

a(n) = SUM[i=1..n] A000069(i) = SUM[i=1..n] {i such that A010060(i)=1}.

a(n) = n^2 - n/2 + O(1). - Charles R Greathouse IV, Mar 22 2013

A177801 Record lengths of chains of consecutive evil primes, starting with A177748(n).

Original entry on oeis.org

2, 3, 5, 7, 8, 12, 16, 20, 23, 25, 26, 30, 31, 32, 34, 38, 39, 40, 41, 42, 44

Offset: 1

Vladimir Shevelev, Dec 12 2010

In contrast to the sequence of all positive integers, where the length of every chain of consecutive evil numbers cannot exceed 2, we conjecture that for the sequence of primes such length is not bounded with growth of n.

Cf. A177800 (odious version), A177748, A177798, A000069, A001969, A027697, A027699.

{l=0;r=0; forprime( p=1, default(primelimit), if( bittest( norml2(binary(p)),0), l>r & print1(r=l ", "); l & l=0, l++))} \\ M. F. Hasler, Dec 12 2010

Extended by D. S. McNeil, Dec 12 2010

a(18)-a(21) from Amiram Eldar, Dec 09 2020

A227921 Odd odious numbers (A000069), all divisors of which are odious.

Original entry on oeis.org

1, 7, 11, 13, 19, 31, 37, 41, 47, 49, 59, 61, 67, 73, 79, 91, 97, 103, 107, 109, 121, 127, 131, 133, 137, 143, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 217, 223, 227, 229, 233, 239, 241, 247, 251, 259, 271, 283, 307, 313, 331, 341, 361, 367, 379, 397, 403

Offset: 1

Vladimir Shevelev, Oct 09 2013

All primes are in A027697.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000069, A027697.

odious:= proc(n) option remember;
  n::odd xor procname(floor(n/2))
end proc:
odious(0):= false:
odious(1):= true:
filter:= proc(n) andmap(odious, numtheory:-divisors(n)) end proc:
select(filter, [seq(i,i=1..500,2)]); # Robert Israel, Aug 18 2019

odiusQ[n_]:=OddQ[Total[IntegerDigits[n,2]]]; Select[Range[1,411,2], odiusQ[ #]&&AllTrue[Divisors[#],odiusQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 08 2019 *)

isodious(n) = {b = binary(n); sum(i=1, #b, b[i]==1) % 2;}
isok(n) = {if (!(n % 2), return (0)); fordiv(n, div, if (! isodious(div), return (0))); return (1);} \\ Michel Marcus, Oct 12 2013

A235985 Primes p such that 3p-1 has even Hamming weight.

Original entry on oeis.org

2, 7, 23, 29, 31, 71, 103, 107, 109, 113, 127, 151, 157, 167, 199, 227, 229, 233, 263, 283, 313, 347, 349, 359, 367, 373, 379, 383, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 541, 569, 599, 607, 619, 631, 647, 739, 761, 797

Offset: 1

Juri-Stepan Gerasimov, Jan 17 2014

Primes p such that A000120(3p-1) is even.

Smallest prime p such that A000120(np-1) is even: 7, 2, 2, 7, 5, 3, 3, 2, 2, 3, 5, 2, 7,...

			23 is in this sequence because A000120(3*23-1) = A000120(68) = 2 (even number).
29 is in this sequence because A000120(3*29-1) = A000120(86) = 4 (even number).

Cf. A019434 (odd primes p having Hamming weight 2), A027697 (primes p having odd Hamming weight), A027699 (primes p having an even Hamming weight).

Select[Prime@Range@200, EvenQ@ First@ DigitCount[3#-1, 2] &] (* Giovanni Resta, Jan 26 2014 *)

isok(p) = isprime(p) && !(hammingweight(3*p-1) % 2); \\ Michel Marcus, Jan 18 2014

A237499 Odious Mersenne exponents.

Original entry on oeis.org

2, 7, 13, 19, 31, 61, 107, 127, 521, 607, 1279, 3217, 11213, 21701, 44497, 132049, 216091, 756839, 1257787, 3021377, 6972593, 20996011, 24036583, 30402457, 37156667, 43112609, 82589933

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Feb 08 2014

Intersection of A000043 and A027697.

Cf. A000069, A174265.

Select[MersennePrimeExponent[Range[47]], OddQ @ DigitCount[#, 2][[1]] &] (* Amiram Eldar, Dec 10 2019 *)

a(13) and a(18) inserted and more terms added by Amiram Eldar, Dec 10 2019

a(27) added by Harvey P. Dale, Feb 01 2025

A269458 Primes p such that the numbers of negabinary evil primes and negabinary odious primes not exceeding p are equal (see comment).

Original entry on oeis.org

3, 53, 61, 71, 89, 101, 107, 7121, 7129, 7159, 7187, 424891, 29739371, 29740511, 29740523, 29740723, 1844046469, 1844046481, 1844046517, 1844046571, 1844046629, 1844046679, 1844046733, 1844046793, 1844046851, 1844047357, 1844047421, 1844047501, 1845540199, 1847154073, 1847154109

Offset: 1

Vladimir Shevelev, Feb 27 2016

Negabinary evil and odious primes are primes in A268272 and A268273 correspondingly.
They are 2,5,7,13,17,19,31,37,61,67,73,79,...
and 3,11,23,29,41,43,47,53,59,71,83,89,...
In contrast to the analogous sequences 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.

Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Negabinary

Cf. A039724, A269027, A027697, A027699, A268483.

aQ[n_] := EvenQ @ Total @ Rest @ Reverse @ Mod[NestWhileList[(# - Mod[#, 2])/-2 &, n, # != 0 &], 2]; s = {}; p = 2; c = 0; Do[If[aQ[p], c++, c--]; If[c == 0, AppendTo[s, p]]; p = NextPrime[p], {10^3}]; s (* Amiram Eldar, Sep 22 2019 after Michael De Vlieger at A268272 *)

More terms from Peter J. C. Moses, Feb 27 2016

More terms from Amiram Eldar, Sep 22 2019

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

A230120 a(n) is the number of evil integers (A001969) not exceeding n and prime to n.

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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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A127977 The minimum excess in the prime race of odious primes versus evil primes in the interval (2^(n-1),2^n).

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A173209 Partial sums of A000069.

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A177801 Record lengths of chains of consecutive evil primes, starting with A177748(n).

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A227921 Odd odious numbers (A000069), all divisors of which are odious.

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A235985 Primes p such that 3p-1 has even Hamming weight.

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A237499 Odious Mersenne exponents.

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