A235001 -id:A235001 - cp's OEIS frontend

A295298 Numbers n for which sum of the divisors (A000203) and the binary weight of n (A000120) have the same parity.

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 23, 24, 25, 27, 29, 30, 32, 33, 34, 39, 40, 43, 45, 46, 48, 49, 50, 51, 53, 54, 57, 58, 60, 63, 64, 65, 66, 68, 71, 75, 77, 78, 80, 81, 83, 85, 86, 89, 90, 92, 95, 96, 98, 99, 100, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 121, 123, 125, 126, 128, 129, 130, 132, 135, 136

Offset: 1

Antti Karttunen, Nov 26 2017

nonn

Numbers n for which A010060(n) = A053866(n).
This sequence is the union of all terms of A028982 (squares and twice squares) that are odious (A000069), and all evil numbers (A001969) that are neither a square or twice a square. See also A231431, A235001.
Sequence A003401 is a subsequence of this sequence. This follows because the only terms in A003401 that are squares or twice squares are the powers of 2 (A000079, that have just one 1-bit, thus are odious), while all the other terms (obtained by multiplying distinct Fermat primes possibly with some power of 2) have an even number of 1-bits, and certainly cannot be squares nor twice squares. - Antti Karttunen, Nov 27 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Positions of zeros in A295297.
Complement of A295299.
Cf. A000069, A001969, A000120, A000230, A010060, A028982, A053866, A231431, A235001.
Cf. A000079, A003401, A295296 (subsequences), also A191363 (the five known terms).

Select[Range@ 136, SameQ @@ Map[EvenQ, {DivisorSigma[1, #], DigitCount[#, 2, 1]}] &] (* Michael De Vlieger, Nov 26 2017 *)

A295299 Numbers k such that the sum of the divisors (A000203) and the binary weight of k (A000120) have different parity.

Original entry on oeis.org

7, 9, 11, 13, 14, 18, 19, 21, 22, 26, 28, 31, 35, 36, 37, 38, 41, 42, 44, 47, 52, 55, 56, 59, 61, 62, 67, 69, 70, 72, 73, 74, 76, 79, 82, 84, 87, 88, 91, 93, 94, 97, 103, 104, 107, 109, 110, 112, 115, 117, 118, 122, 124, 127, 131, 133, 134, 137, 138, 140, 143, 144, 145, 146, 148, 151, 152, 155, 157, 158, 161, 164, 167, 168, 169

Offset: 1

Antti Karttunen, Nov 26 2017

nonn

Numbers k for which A010059(k) = A053866(k).
This sequence is the union of all terms of A028982 (squares and twice squares) that are evil (A001969), and all odious numbers (A000069) that are neither a square or twice a square. See also A231431, A235001.
This is a subsequence of A004169, numbers k such that phi(k) is not a power of 2. See comment in A295298 for the reason. - Antti Karttunen, Nov 27 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Complement of A295298.
Subsequence of A004169.
Cf. A000069, A001969, A000120, A000203, A010059, A028982, A053866, A231431, A235001, A295297 (characteristic function).

Select[Range@ 169, UnsameQ @@ Map[EvenQ, {DivisorSigma[1, #], DigitCount[#, 2, 1]}] &] (* Michael De Vlieger, Nov 26 2017 *)

A235331 Numbers with odious squares.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 18, 19, 20, 22, 23, 25, 28, 29, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 49, 50, 51, 55, 56, 58, 59, 62, 64, 65, 66, 67, 68, 70, 71, 72, 75, 76, 77, 78, 79, 80, 82, 85, 86, 87, 88, 89, 91, 92, 95, 97, 98, 100

Offset: 1

Gerasimov Sergey, Jan 06 2014

Numbers that are sqrt(odious squares).

			1 = sqrt(A235001(1)) = sqrt(1).
2 = sqrt(A235001(2)) = sqrt(4).
4 = sqrt(A235001(3)) = sqrt(16).
5 = sqrt(A235001(4)) = sqrt(25).

Vincenzo Librandi, Table of n, a(n) for n = 1..2600

Cf. A235001, A194991.

[n : n in [0..130] |IsOdd(&+Intseq(n^2, 2))]; // Vincenzo Librandi, Jan 31 2018

Select[Range[200], OddQ[DigitCount[#^2, 2][[1]]] &] (* Vincenzo Librandi, Jan 31 2018 *)

isok(n) = (hammingweight(n^2) % 2) == 1; \\ Michel Marcus, Jan 31 2018

a(n) = sqrt(A235001(n)).

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");

A234037 The union of odious numbers with evil squares and evil numbers with odious squares.

Original entry on oeis.org

5, 9, 10, 13, 17, 18, 20, 21, 23, 26, 29, 33, 34, 36, 37, 39, 40, 42, 43, 46, 47, 51, 52, 58, 61, 65, 66, 68, 69, 71, 72, 73, 74, 75, 77, 78, 80, 81, 84, 85, 86, 89, 92, 93, 94, 95, 101, 102, 104, 107, 109, 113, 115, 116, 122, 125, 129, 130, 132, 133, 135, 136, 137

Offset: 1

Gerasimov Sergey, Jan 13 2014

Numbers n with odd sum of binary weight of n and binary weight of n^2.
Primes are in this sequence: 5, 13, 17, 23, 29, 37, 43, 47, 61, 71, 73,....
Evil numbers with odious squares: 5, 9, 10, 17, 18, 20, 23, 29, 33, 34,...
Odious numbers with evil squares: 13, 21, 26, 37, 42, 47, 52, 61, 69,...

			a(3) = 10 because 2 (binary weight of 10) + 3 (binary weight of 100) = 5 (odd).
a(4) = 13 because 3 (binary weight of 13) + 4 (binary weight of 169) = 7 (odd).

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

Cf. A000069, A000120, A001969, A159918, A231431, A235001.

Select[Range[100], Mod[DigitCount[#, 2, 1], 2] != Mod[DigitCount[#^2, 2, 1], 2] &] (* Amiram Eldar, Aug 31 2020 *)

Wrong term removed by Amiram Eldar, Aug 31 2020

A298879 Numbers whose square is not odious.

Original entry on oeis.org

0, 3, 6, 12, 13, 15, 21, 24, 26, 27, 30, 37, 42, 45, 47, 48, 52, 53, 54, 57, 60, 61, 63, 69, 73, 74, 81, 83, 84, 90, 93, 94, 96, 99, 104, 105, 106, 107, 108, 109, 111, 114, 115, 119, 120, 122, 123, 126, 133, 137, 138, 141, 146, 148, 151, 155, 159

Offset: 1

Vincenzo Librandi, Jan 31 2018

Complement of A235331.
From Robert Israel, Feb 02 2018: (Start)
2*n is in the sequence if and only if n is in the sequence.
2*n+1 is in the sequence if and only if n*(n+1) is odious. (End)

Cf. A235001, A235331.

[n: n in [0..200] | IsEven(&+Intseq(n^2, 2))];

select(t -> convert(convert(t^2,base,2),`+`)::even, [$0..200]); # Robert Israel, Feb 02 2018

Join[{0}, Select[Range[200], !OddQ[DigitCount[#^2, 2][[1]]] &]]

isok(n) = (hammingweight(n^2) % 2) != 1; \\ Michel Marcus, Jan 31 2018

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A295298 Numbers n for which sum of the divisors (A000203) and the binary weight of n (A000120) have the same parity.

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A295299 Numbers k such that the sum of the divisors (A000203) and the binary weight of k (A000120) have different parity.

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A235331 Numbers with odious squares.

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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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A234037 The union of odious numbers with evil squares and evil numbers with odious squares.

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A298879 Numbers whose square is not odious.

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