A047476 -id:A047476 - cp's OEIS frontend

A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

2, 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 547, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 659, 673, 683

Offset: 1

N. J. A. Sloane

Sequence naturally partitions into two sequences: all primes p with ord_p(-2) odd (A163183, the primes dividing 2^j +1 for some odd j) and certain primes p with ord_p(-2) even (A163185). - Christopher J. Smyth, Jul 23 2009

Terms m in A047476 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002332, A002333.
Cf. A039706, A003628 (complement with respect to A000040).
Primes in A002479.
Cf. A051100 (see Mathar's comment).
Apart from leading term the same as A033200.

a033203 n = a033203_list !! (n-1)
a033203_list = filter ((== 1) . a010051) a047476_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1..3]]; // Vincenzo Librandi, Aug 11 2012

[p: p in PrimesUpTo(800) | NormEquation(2,p) eq true]; // Bruno Berselli, Jul 03 2016

QuadPrimes2[1, 0, 2, 10000] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,8]]&] (* Harvey P. Dale, Mar 16 2013 *)

is(n)=isprime(n) && issquare(Mod(-2,n)) \\ Charles R Greathouse IV, Nov 29 2016

a(n) = A002332(n) + 2*A002333(n)^2. - Zak Seidov, May 29 2014

A103127 Numbers congruent to {-1, 1, 3, 5} mod 16.

Original entry on oeis.org

1, 3, 5, 15, 17, 19, 21, 31, 33, 35, 37, 47, 49, 51, 53, 63, 65, 67, 69, 79, 81, 83, 85, 95, 97, 99, 101, 111, 113, 115, 117, 127, 129, 131, 133, 143, 145, 147, 149, 159, 161, 163, 165, 175, 177, 179, 181, 191, 193, 195, 197, 207, 209, 211, 213, 223, 225, 227, 229, 239, 241

Offset: 1

N. J. A. Sloane, Mar 25 2005

Agrees with A103192 for the first 511 terms, but then diverges (see comment in A103192). - Bruno Berselli, Dec 01 2016

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp [pdf, ps].
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences which agree for a long time but are different

Cf. A047527, A103192.

a103127 n = a103127_list !! (n-1)
a103127_list = [x | x <- [1..], x `mod` 16 `elem` [1,3,5,15]]
-- Reinhard Zumkeller, Jul 21 2012

Select[Range[300],MemberQ[{1,3,5,15},Mod[#,16]]&] (* Harvey P. Dale, Aug 10 2019 *)

a(n) = 2*A047527(n) + 1.
From R. J. Mathar, Aug 30 2008: (Start)
O.g.f.: x*(1 + 2*x + 2*x^2 + 10*x^3 + x^4)/((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-4) + 16. (End)
a(n) = 2*A047476(n+1) - 1. - Philippe Deléham, Dec 01 2016

A293292 Numbers with last digit less than 5 (in base 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 80, 81, 82, 83, 84, 90, 91, 92, 93, 94, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130

Offset: 1

Bruno Berselli, Oct 05 2017

Equivalently, numbers k such that floor(k/5) = 2*floor(k/10).
After 0, partial sums of A010122 starting from the 2nd term.
The sequence differs from A007091 after a(25).
Also numbers k such that floor(k/5) is even. - Peter Luschny, Oct 05 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A010122, A239229, A257145, A293481 (complement).
Sequences of the type floor(n/d) = (10/d)*floor(n/10), where d is a factor of 10: A008592 (d=1), A197652 (d=2), this sequence (d=5), A001477 (d=10).
Sequences of the type n + r*floor(n/r): A005843 (r=1), A042948 (r=2), A047240 (r=3), A047476 (r=4), this sequence (r=5).

Magma
```
[n: n in [0..130] | n mod 10 lt 5];
```

[n: n in [0..130] | IsEven(Floor(n/5))];

Magma
```
[n+5*Floor(n/5): n in [0..70]];
```

select(k -> type(floor(k/5), even), [$0..130]); # Peter Luschny, Oct 05 2017

Table[n + 5 Floor[n/5], {n, 0, 70}]
Reap[For[k = 0, k <= 130, k++, If[Floor[k/5] == 2*Floor[k/10], Sow[k]]]][[2, 1]] (* or *) LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 10}, 66] (* Jean-François Alcover, Oct 05 2017 *)

concat(0, Vec(x^2*(1 + x + x^2 + x^3 + 6*x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^70))) \\ Colin Barker, Oct 05 2017

select(k->floor(k/5) == 2*floor(k/10), vector(1000, k, k)) \\ Colin Barker, Oct 05 2017

[k for k in range(131) if (k//5) % 2 == 0] # Peter Luschny, Oct 05 2017

def A293292(n): return (n-1<<1)-(n-1)%5 # Chai Wah Wu, Oct 29 2024

[k for k in (0..130) if 2.divides(floor(k/5))] # Peter Luschny, Oct 05 2017

G.f.: x^2*(1 + x + x^2 + x^3 + 6*x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(n) = (n-1) + 5*floor((n-1)/5) = 10*floor((n-1)/5) + ((n-1) mod 5).
a(n) = A257145(n+2) - A239229(n-1). - R. J. Mathar, Oct 05 2017
a(n) = 2n-2-((n-1) mod 5). - Chai Wah Wu, Oct 29 2024

Definition by David A. Corneth, Oct 05 2017

A295653 Square array T(n, k), n >= 0, k >= 0, read by antidiagonals upwards: T(n, k) = the (k+1)-th nonnegative number m such that n AND m = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 1, 4, 3, 0, 4, 4, 6, 4, 0, 1, 8, 5, 8, 5, 0, 2, 2, 12, 8, 10, 6, 0, 1, 8, 3, 16, 9, 12, 7, 0, 8, 8, 10, 8, 20, 12, 14, 8, 0, 1, 16, 9, 16, 9, 24, 13, 16, 9, 0, 2, 2, 24, 16, 18, 10, 28, 16, 18, 10, 0, 1, 4, 3, 32, 17, 24, 11, 32, 17, 20

Offset: 0

Rémy Sigrist, Nov 25 2017

This sequence has similarities with A126572: here we check for common bits in binary representations, there for common primes in prime factorizations.
For any n >= 0 and k >= 0:
- T(0, k) = k,
- T(1, k) = 2*k,
- T(2, k) = A042948(k),
- T(3, k) = 4*k,
- T(4, k) = A047476(k),
- T(5, k) = A047467(k),
- T(2^n - 1, k) = 2^n * k,
- T(n, 0) = 0,
- T(n, 1) = A006519(n+1),
- T(n, k + 2^A080791(n)) = T(n, k) + 2^A029837(n+1) (i.e. each row is linear),
- A000120(T(n, k)) = A000120(k).

			Square array begins:
n\k  0   1   2   3   4   5   6   7   8   9  ...
0:   0   1   2   3   4   5   6   7   8   9  ...
1:   0   2   4   6   8  10  12  14  16  18  ...
2:   0   1   4   5   8   9  12  13  16  17  ...
3:   0   4   8  12  16  20  24  28  32  36  ...
4:   0   1   2   3   8   9  10  11  16  17  ...
5:   0   2   8  10  16  18  24  26  32  34  ...
6:   0   1   8   9  16  17  24  25  32  33  ...
7:   0   8  16  24  32  40  48  56  64  72  ...
8:   0   1   2   3   4   5   6   7  16  17  ...
9:   0   2   4   6  16  18  20  22  32  34  ...

Cf. A000120, A006519, A029837, A042948, A047467, A047476, A080791, A126572.

T(n,k) = if (n==0, k, n%2, 2*T(n\2,k), 2*T(n\2,k\2) + (k%2))

For any n >= 0 and k >= 0:
- T(0, k) = k,
- T(2*n + 1, k) = 2*T(n, k),
- T(2*n, 2*k) = 2*T(n, k),
- T(2*n, 2*k + 1) = 2*T(n, k) + 1.
For any n >= 0, T(n, k) ~ 2^A000120(n) * k as k tends to infinity.

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A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

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A103127 Numbers congruent to {-1, 1, 3, 5} mod 16.

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A293292 Numbers with last digit less than 5 (in base 10).

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A295653 Square array T(n, k), n >= 0, k >= 0, read by antidiagonals upwards: T(n, k) = the (k+1)-th nonnegative number m such that n AND m = 0 (where AND denotes the bitwise AND operator).

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