A027697 -id:A027697 - cp's OEIS frontend

A357761 a(n) = A227872(n) - A356018(n).

1, 2, 0, 3, 0, 0, 2, 4, -1, 0, 2, 0, 2, 4, -2, 5, 0, -2, 2, 0, 2, 4, 0, 0, 1, 4, -2, 6, 0, -4, 2, 6, 0, 0, 2, -3, 2, 4, 0, 0, 2, 4, 0, 6, -4, 0, 2, 0, 3, 2, -2, 6, 0, -4, 2, 8, 0, 0, 2, -6, 2, 4, 0, 7, 0, 0, 2, 0, 0, 4, 0, -4, 2, 4, -2, 6, 2, 0, 2, 0, -1, 4, 0

Offset: 1

Amiram Eldar, Oct 12 2022

The excess of the number of odious (A000069) divisors of n over the number of evil (A001969) divisors of n.

Every integer occurs in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000069, A000290 (positions of odd terms), A001969, A027697, A027699, A106400, A227872, A230851 (positions of 0's), A356018, A357762.

Similar sequences: A046660, A048272.

a[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; Array[a, 100]

a(n) = -sumdiv(n, d, (-1)^hammingweight(d));

a(n) = -Sum_{d|n} A106400(d).
a(n) = A000005(n) - 2*A356018(n).
a(n) = 2*A227872(n) - A000005(n).
a(n) = 0 iff n is in A230851.
a(n) == 1 (mod 2) iff n is a square (A000290).
a(2^n) = n + 1.
a(p*2^n) = 0 when p is an evil prime (A027699).
a(p^2*2^n) = n + 1 when p is an evil prime (A027699) and p^2 is odious, and when p is an odd odious prime (A027697) and p^2 is evil.
a(p^2*2^n) = -(n+1) when p is an evil prime and p^2 is also evil.
a(p^2*2^n) = 3*(n+1) when p is an odd odious prime and p^2 is also odious.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Sum_{k>=1} A106400(k)/k = 1.196283264... (A357762).

A177748 First primes of record chains of consecutive primes such that all of them are evil (A027699).

Original entry on oeis.org

3, 257, 337, 4423, 4919, 30431, 66841, 514271, 14490383, 231234569, 325923613, 640085473, 1600993259, 7180164577, 8069913503, 86933359951, 284331217637, 1128352801153, 1209935587291, 2454267258251, 2945783287813

Offset: 1

Vladimir Shevelev, Dec 12 2010

The first lengths of such chains of primes are: 2, 3, 5, 7, 8, 12, 16, ..., cf. A177801.

			257 is the first evil prime followed by two consecutive primes (263,269) which are also evil. Thus it is record length 3.

Cf. A001969, A000069, A027697, A027699, A177798 (odious version), A177801.

{l=0;sp=0;r=0; forprime( p=1, default(primelimit), if( norml2(binary(p))%2, l>r & !print1(sp", ") & r=l; l & l=0, l || sp=p; l++))} \\ M. F. Hasler, Dec 12 2010

More terms from D. S. McNeil, Dec 12 2010

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

A194991 Primes whose squares are odious.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 59, 67, 71, 79, 89, 97, 101, 103, 113, 127, 131, 139, 149, 157, 163, 167, 173, 179, 181, 193, 197, 223, 227, 229, 239, 251, 257, 263, 271, 283, 307, 313, 349, 353, 373, 379, 383, 389, 401, 409, 421, 431, 433, 439, 449, 457, 467, 479, 487, 509, 523, 547, 563

Offset: 1

Vladimir Shevelev, Sep 07 2011

Subsequence of the numbers 1, 2, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 18, 19, 20, 22, 23, 25, 28, 29, 31, 32, 33, ... which have odious squares. See A235331. - R. J. Mathar, Sep 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..5240

Cf. A000069, A001969, A027697, A027699.

[p : p in PrimesInInterval(2, 600) |IsOdd(&+Intseq(p^2, 2))]; // Vincenzo Librandi, Dec 15 2018

Select[Prime@ Range@ 120, OddQ@ First@ DigitCount[#^2, 2] &] (* Amiram Eldar, Dec 14 2018 after Michael De Vlieger at A027697 *)

isok(p) = isprime(p) && (hammingweight(p^2) % 2); \\ Michel Marcus, Dec 14 2018

A227930 Primes p such that p-1 and p+1 have an even Hamming weight.

Original entry on oeis.org

11, 19, 47, 59, 67, 79, 107, 131, 179, 191, 211, 227, 239, 251, 271, 283, 307, 331, 367, 379, 419, 431, 443, 463, 491, 499, 563, 587, 659, 719, 787, 827, 859, 883, 911, 947, 971, 1019, 1039, 1051, 1087, 1123, 1163, 1171, 1187, 1231, 1259, 1279, 1291, 1327, 1423, 1451, 1459, 1471, 1483

Offset: 1

Juri-Stepan Gerasimov, Oct 06 2013

Primes such that both neighbors are evil (as defined in A001969).
From Antti Karttunen, Dec 29 2013: (Start)
Excluding 2, the intersection of A027697 (Odious primes: primes with odd number of 1's in binary expansion) and A095282 (Primes whose binary-expansion ends with an even number of 1's).
Equally, the intersection of A092246 (Odd "odious" numbers) and A095282.
Equally, odd odious primes p such that A007814(p+1) is even.
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002145.
Cf. A000069, A001969, A007814, A027697, A092246, A095282.
Seems to consist of all primes in A233388.

read("transforms"):
isA000069 := proc(n)
    if wt(n) mod 2 = 1 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 do
    if isprime(n) and not isA000069(n-1) and not isA000069(n+1) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 08 2013
(Scheme, with Antti Karttunen's IntSeq-library, two alternative implementations)
(define A227930 (MATCHING-POS 1 1 (lambda (n) (and (even? (A000120 (- n 1))) (even? (A000120 (+ n 1))) (prime? n)))))
(define A227930v2 (MATCHING-POS 1 1 (lambda (n) (and (odd? n) (odd? (A000120 n)) (even? (A007814 (+ n 1))) (prime? n))))) # Antti Karttunen, Dec 29 2013

Select[Prime[Range[250]], And @@ EvenQ[DigitCount[# + {-1, 1}, 2, 1]] &] (* Amiram Eldar, Jul 24 2023 *)

is(n)=hammingweight(n-1)%2==0 && hammingweight(n+1)%2==0 && isprime(n) \\ Charles R Greathouse IV, Oct 09 2013

Entries checked by R. J. Mathar, Oct 08 2013

A230070 a(n) is the number of odious integers (A000069) not exceeding n and respectively prime to n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 2, 5, 2, 5, 3, 6, 3, 8, 4, 9, 4, 9, 5, 8, 5, 12, 5, 12, 6, 13, 5, 15, 5, 15, 8, 14, 8, 12, 8, 18, 9, 17, 8, 20, 8, 22, 10, 19, 11, 23, 11, 18, 11, 24, 12, 27, 12, 21, 10, 25, 14, 29, 11, 30, 15, 24, 16, 26, 13, 33, 17, 32, 12, 36, 16, 36

Offset: 1

Vladimir Shevelev, Oct 10 2013

Let b(n) is the number of evil integers (A001969) not exceeding n and respectively prime to n. Then a(n) + b(n) = phi(n) (phi = A000010). For which numbers a(n) < b(n)? This sequence begins 28,... . For n = 1,2,3,15, we have a(n) = phi(n). What other solutions has this equation? When a(n) = phi(n)/2, we call n a balanced number. The sequence of balanced numbers begins 4,6,7,8,10,11,13,14,16,19,22,...

			For n = 30, we have the following numbers respectively prime to n: 1, 7, 11, 13, 17, 19, 23, 29, from which only 5 numbers 1, 7, 11, 13 and 19 are odious. So, a(30) = 5.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A000010, A000069, A001969, A027697, A027699.

odiouses=Select[Range[rng=100],OddQ[DigitCount[#,2][[1]]]&]; tmp=1; Table[Count[Map[CoprimeQ[n,#]&, Take[odiouses, tmp=NestWhile[#+1&,tmp+1, odiouses[[#]]

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

For odd prime p, a(p) = (p + 1 or - 1)/2. Primes p for which a(p) = (p+1)/2 are 3, 5, 17, 23, 29,..., i.e., evil primes (A027699), while odd primes p for which a(p) = (p-1)/2 are 7,11,13,19,..., i.e., odious primes (A027697).

A230095 Odious numbers (A000069) that are the product of exactly two distinct primes.

Original entry on oeis.org

14, 21, 22, 26, 35, 38, 55, 62, 69, 74, 82, 87, 91, 93, 94, 115, 118, 122, 133, 134, 143, 145, 146, 155, 158, 161, 185, 194, 203, 205, 206, 213, 214, 217, 218, 247, 253, 254, 259, 262, 265, 274, 295, 299, 301, 302, 309, 314, 319, 321, 327, 334, 339, 341, 346

Offset: 1

N. J. A. Sloane, Oct 11 2013

Charles R Greathouse IV, 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).

Cf. A000069, A027697, A130593.

Select[Range[400],OddQ[DigitCount[#,2,1]]&&PrimeNu[#]==PrimeOmega[#]==2&] (* Harvey P. Dale, May 21 2024 *)

isodious(n)=b = binary(n); sum(i=1, #b, b[i]==1) % 2;
isok(n) = isodious(n) && (bigomega(n)==2) && (omega(n)==2); \\ Michel Marcus, Oct 12 2013

list(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p-1), if(hammingweight(p*q)%2, listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

More terms from Michel Marcus, Oct 12 2013

A232667 Primes p such that the p-th odious number is prime; odious primes p such that 2p-1 is prime.

Original entry on oeis.org

2, 7, 19, 31, 37, 79, 97, 157, 199, 211, 229, 271, 307, 331, 367, 379, 439, 499, 577, 601, 607, 661, 727, 829, 877, 967, 997, 1009, 1069, 1171, 1279, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011, 2029, 2131, 2137, 2311, 2551, 2557, 3037, 3061, 3109, 3169, 3181

Offset: 1

Juri-Stepan Gerasimov, Nov 27 2013

nonn

From Antti Karttunen, Nov 29 & 30 2013: (Start)
This sequence is the intersection of A005382 and A027697.
Proof:
A000069(n) reduces according to the bit parity of n-1 as follows:
  A000069(n) = 2n - 2 when n-1 is odious.
  A000069(n) = 2n - 1 when n-1 is evil.
which means that no prime in this sequence can be evil, as then p-1 would be an odious number (true for all odd primes) and A000069(p) would be 2(p-1) which obviously cannot be a prime, contradicting the requirement. Thus all primes present must belong to the set of odious primes, A027697.
As each prime p here is thus odious, it means that each p-1 is an evil number (A001969), and thus A000069(p) = 2p-1. And the stipulation that it also must be prime, is just what is required from the terms of A005382. Thus this sequence contains exactly those primes that occur in both A005382 and A027697.
Equally: this is the intersection of A000069 and A005382, thus prime p occurs here iff A000120(p) is odd and 2p-1 is prime also.
Also, apart from the first term (2), all the primes (2*a(n))-1 are also odious. This follows because for any odd number k, A000120(2k-1) = A000120(k).
(End)

			7 is a prime and A000069(7) = 13, a prime also, thus 7 is in this sequence.
19 is a prime and A000069(19) = 37, a prime also, thus 19 is in this sequence.
Alternatively:
7 is a prime, 2*7-1 = 13 is also prime, and when written in binary, 7 = '111', with an odd number of 1-bits. Thus 7 is included in this sequence.
The next time this happens, is for 19, as it is a prime, 2*19-1 = 37 is also prime, and when written in binary, 19 = '10011', also has on odd number of 1-bits.

Antti Karttunen, Table of n, a(n) for n = 1..225

Cf. A000040, A000069, A001969, A000120, A005382, A027697, A092246, A232637.

Edited and erroneous terms removed by Antti Karttunen, Nov 29-30 2013

A291762 Restricted growth sequence transform of ((-1)^A000120(n))*A046523(n); filter combining the parity of binary weight with the prime signature of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Equally, restricted growth sequence transform of sequence b defined as b(1) = 1; b(n) = A046523(n) + A010060(n) for n > 1, which starts as 1, 3, 2, 5, 2, 6, 3, 9, 4, 6, 3, 12, 3, 7, 6, 17, 2, 12, 3, 12, 7, 7, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000120, A010060, A046523.
Cf. A101296, A286163, A291761 (related or similar filtering sequences).
Cf. A027697 (positions of 2's), A027699 (of 3's), A130593 (of 5's and 7's), A230095 (of 9's).
Cf. also A231431, A235001.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
write_to_bfile(1,rgs_transform(vector(65537,n,((-1)^hammingweight(n))*A046523(n))),"b291762_upto65537.txt");
\\ Or alternatively:
A010060(n) = (hammingweight(n)%2);
f(n) = if(1==n,n,A046523(n)+A010060(n));
write_to_bfile(1,rgs_transform(vector(16385,n,f(n))),"b291762.txt");

A176620 Primes p for which the factorization of p! over distinct terms of A050376 does not contain 2.

Original entry on oeis.org

7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443, 457, 463, 487

Offset: 1

Vladimir Shevelev, Apr 22 2010

nonn

Equivalent definition: primes p for which A007814(p!) is even. Apparently, the sequence is A027697 without the 2 (see A014499). [R. J. Mathar, Oct 26 2010]

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050376, A000040, A000142.

Select[Range[500], PrimeQ[#] && EvenQ @ IntegerExponent[#!, 2] &]  (* Amiram Eldar, Sep 13 2019 *)

Corrected (37 added, 41 added, 43 removed...) and extended by R. J. Mathar, Oct 26 2010

A177800 The maximal length of the chain of consecutive primes starting with A177798(n) such that all of them are odious.

Original entry on oeis.org

1, 3, 6, 9, 11, 15, 17, 18, 22, 23, 25, 26, 28, 29, 33, 34, 35, 36, 38, 39, 42

Offset: 1

Vladimir Shevelev, Dec 12 2010

In contrast to sequence of all positive integers, where the length of a chain of consecutive odious numbers cannot exceed 2, we conjecture that over primes the length is not bounded.

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

Extended by D. S. McNeil, Dec 12 2010

a(20)-a(21) from Amiram Eldar, Dec 09 2020

cp's OEIS Frontend

A357761 a(n) = A227872(n) - A356018(n).

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A177748 First primes of record chains of consecutive primes such that all of them are evil (A027699).

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A194991 Primes whose squares are odious.

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Magma

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A227930 Primes p such that p-1 and p+1 have an even Hamming weight.

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Maple

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A230070 a(n) is the number of odious integers (A000069) not exceeding n and respectively prime to n.

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A230095 Odious numbers (A000069) that are the product of exactly two distinct primes.

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A232667 Primes p such that the p-th odious number is prime; odious primes p such that 2p-1 is prime.

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A291762 Restricted growth sequence transform of ((-1)^A000120(n))*A046523(n); filter combining the parity of binary weight with the prime signature of n.

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