A047621 -id:A047621 - cp's OEIS frontend

A306220 a(n) is the smallest prime p such that Kronecker(-n,p) = -1.

3, 5, 2, 3, 2, 13, 3, 5, 7, 3, 2, 5, 2, 11, 7, 3, 5, 5, 2, 11, 2, 3, 5, 13, 3, 11, 2, 3, 2, 7, 3, 5, 5, 3, 2, 7, 2, 5, 7, 3, 13, 5, 2, 7, 2, 3, 5, 5, 3, 7, 2, 3, 2, 13, 3, 11, 5, 3, 2, 7, 2, 5, 5, 3, 7, 19, 2, 5, 2, 3, 7, 5, 3, 7, 2, 3, 2, 5, 3, 11, 7, 3, 2, 13, 2, 7, 5

Offset: 1

Jianing Song, Jan 29 2019

nonn

A companion sequence to A092419.

Conjecture: lim sup log(a(n))/log(n) = 0. For example, it seems that log(a(n))/log(n) < 0.5 for all n > 1364.

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

Cf. A092419.

See A306224 for another version.

# This requires Maple 2016 or later
f:= proc(n) local p;
  p:= 2;
  while NumberTheory:-KroneckerSymbol(-n,p) <> -1 do p:= nextprime(p) od:
  p
end proc:
map(f, [$1..100]); # Robert Israel, Feb 17 2019

a[n_] := For[p = 2, True, p = NextPrime[p], If[KroneckerSymbol[-n, p] == -1, Return[p]]];
Array[a, 100] (* Jean-François Alcover, Jun 18 2020 *)

a(n) = forprime(p=2, , if(kronecker(-n, p)<0, return(p)))

a(n) = 2 if and only if n == 3, 5 (mod 8). See A047621.

a(n) = 3 if and only if n == 1, 4, 7, 10, 16, 22 (mod 24).

A047596 Numbers that are congruent to {2, 3, 4, 5} mod 8.

Original entry on oeis.org

2, 3, 4, 5, 10, 11, 12, 13, 18, 19, 20, 21, 26, 27, 28, 29, 34, 35, 36, 37, 42, 43, 44, 45, 50, 51, 52, 53, 58, 59, 60, 61, 66, 67, 68, 69, 74, 75, 76, 77, 82, 83, 84, 85, 90, 91, 92, 93, 98, 99, 100, 101, 106, 107, 108, 109, 114, 115, 116, 117, 122, 123

Offset: 1

N. J. A. Sloane

For n > 2, a(n) is the decimal value that results from the conversion of n-1 to binary whose last two bits are altered by either of the following rules: 00->010, 01->011, 10->100, 11->101. For example a(10) = 19 because 10 - 1 = 9 = '10'01'->'10'011' = 19. - Franck Maminirina Ramaharo, Jul 25 2018

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

Cf. A010873, A047463, A047621.

Filtered([1..130],n->n mod 8=2 or n mod 8=3 or n mod 8=4 or n mod 8=5); # Muniru A Asiru, Jul 27 2018

[n: n in [1..120] | n mod 8 in [2..5]]; // Bruno Berselli, Jul 17 2012

A047596:=n->2*n-1-I^(n*(n+1))-(1+I^(2*n))/2: seq(A047596(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[120], MemberQ[{2, 3, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {2, 3, 4, 5, 10}, 60] (* Bruno Berselli, Jul 17 2012 *)

makelist(2*n-1-%i^(n*(n+1))-(1+(-1)^n)/2,n,1,60); /* Bruno Berselli, Jul 17 2012 */

Vec((2+x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) \\ Bruno Berselli, Jul 17 2012

G.f.: x*(2+x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)). [Bruno Berselli, Jul 17 2012]
a(n) = 2*n-1-i^(n*(n+1))-(1+(-1)^n)/2, where i=sqrt(-1). [Bruno Berselli, Jul 17 2012]
a(n) = 2n - A010873(n-1). - Wesley Ivan Hurt, Jul 07 2013
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047621(k), a(2k-1) = A047463(k). (End)
E.g.f.: 3 + sin(x) - cos(x) + (2*x - 1)*sinh(x) + 2*(x - 1)*cosh(x). - Ilya Gutkovskiy, Jun 02 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - (sqrt(2)+3)*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 25 2021

A047599 Numbers that are congruent to {0, 3, 4, 5} mod 8.

Original entry on oeis.org

0, 3, 4, 5, 8, 11, 12, 13, 16, 19, 20, 21, 24, 27, 28, 29, 32, 35, 36, 37, 40, 43, 44, 45, 48, 51, 52, 53, 56, 59, 60, 61, 64, 67, 68, 69, 72, 75, 76, 77, 80, 83, 84, 85, 88, 91, 92, 93, 96, 99, 100, 101, 104, 107, 108, 109, 112, 115, 116, 117, 120, 123, 124

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A008586, A047621.

[n : n in [0..150] | n mod 8 in [0, 3, 4, 5]]; // Wesley Ivan Hurt, May 22 2016

A047599:=n->2*n-2-(I^(-n)+I^n)/2: seq(A047599(n), n=1..100); # Wesley Ivan Hurt, May 22 2016

Table[2n-2-(I^(-n)+I^n)/2, {n, 80}] (* Wesley Ivan Hurt, May 22 2016 *)
LinearRecurrence[{2,-2,2,-1},{0,3,4,5},80] (* Harvey P. Dale, Mar 27 2023 *)

[lucas_number1(n, 0, 1)+2*n for n in range(0, 55)] # Zerinvary Lajos, Mar 09 2009

From R. J. Mathar, Oct 08 2011: (Start)
G.f.: ( x^2*(3-2*x+3*x^2) ) / ( (x^2+1)*(x-1)^2 ).
a(n) = 2*n-2-cos(n*Pi/2). (End)
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = 2n - 2 - (i^(-n) + i^n)/2 where i = sqrt(-1).
a(2n) = A047621(n), a(2n+1) = A008586(n) for n>0. (End)
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/4 + sqrt(2)*log(3-2*sqrt(2))/8. - Amiram Eldar, Dec 21 2021

More terms from Wesley Ivan Hurt, May 22 2016

A329095 Odd numbers k such that x^2 == 2 (mod k) has no solution.

Original entry on oeis.org

3, 5, 9, 11, 13, 15, 19, 21, 25, 27, 29, 33, 35, 37, 39, 43, 45, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 75, 77, 81, 83, 85, 87, 91, 93, 95, 99, 101, 105, 107, 109, 111, 115, 117, 121, 123, 125, 129, 131, 133, 135, 139, 141, 143, 145, 147, 149, 153, 155, 157, 159, 163

Offset: 1

Jianing Song, Nov 04 2019

Complement of A058529 over the odd numbers: odd numbers k such that x^2 == 2 (mod k) has solutions.
Odd numbers k such that at least one prime factor of k is congruent to 3 or 5 modulo 8 (at least one prime factor is in A003629).
Also odd terms in A025020.

			x^2 == 2 (mod 45) has no solution, so 45 is a term.

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

Cf. A003629. A047621 is a subsequence.

Cf. A058529, A057126, A025020 (numbers k such that x^2 == 2 (mod k) has no solution).

filter:= proc(t) (numtheory:-factorset(t) mod 8) intersect {3,5} <> {} end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Nov 05 2019

Reap[Do[If[AnyTrue[FactorInteger[k][[All, 1]], MatchQ[Mod[#, 8], 3|5]&], Sow[k]], {k, 1, 999, 2}]][[2, 1]] (* Jean-François Alcover, Aug 22 2020 *)

isA329095(k) = (k%2) && !issquare(Mod(2,k))

A047499 Numbers that are congruent to {3, 4, 5, 7} mod 8.

Original entry on oeis.org

3, 4, 5, 7, 11, 12, 13, 15, 19, 20, 21, 23, 27, 28, 29, 31, 35, 36, 37, 39, 43, 44, 45, 47, 51, 52, 53, 55, 59, 60, 61, 63, 67, 68, 69, 71, 75, 76, 77, 79, 83, 84, 85, 87, 91, 92, 93, 95, 99, 100, 101, 103, 107, 108, 109, 111, 115, 116, 117, 119, 123, 124

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047535, A047621.

[n : n in [0..150] | n mod 8 in [3, 4, 5, 7]]; // Wesley Ivan Hurt, May 27 2016

A047499:=n->(8*n-1-I^(2*n)-(1-2*I)*I^(-n)-(1+2*I)*I^n)/4: seq(A047499(n), n=1..100); # Wesley Ivan Hurt, May 27 2016

Table[(8n-1-I^(2n)-(1-2*I)*I^(-n)-(1+2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *)
LinearRecurrence[{1,0,0,1,-1},{3,4,5,7,11},70] (* Harvey P. Dale, May 16 2025 *)

G.f.: x*(3+x+x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Nov 06 2015
From Wesley Ivan Hurt, May 27 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-1-i^(2*n)-(1-2*i)*i^(-n)-(1+2*i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047535(k), a(2k-1) = A047621(k). (End)
E.g.f.: 1 + sin(x) - cos(x)/2 + 2*x*sinh(x) + (2*x - 1/2)*cosh(x). - Ilya Gutkovskiy, May 27 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + (4-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(2-sqrt(2))/8. - Amiram Eldar, Dec 26 2021

A047600 Numbers that are congruent to {1, 3, 4, 5} mod 8.

Original entry on oeis.org

1, 3, 4, 5, 9, 11, 12, 13, 17, 19, 20, 21, 25, 27, 28, 29, 33, 35, 36, 37, 41, 43, 44, 45, 49, 51, 52, 53, 57, 59, 60, 61, 65, 67, 68, 69, 73, 75, 76, 77, 81, 83, 84, 85, 89, 91, 92, 93, 97, 99, 100, 101, 105, 107, 108, 109, 113, 115, 116, 117, 121, 123, 124

Offset: 1

N. J. A. Sloane

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

Cf. A047461, A047621.

[n: n in [1..120] | n mod 8 in [1,3,4,5]];

A047600:=n->2*n-1-(3+I^(2*n))*(1+I^(n*(n+1)))/4: seq(A047600(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[120], MemberQ[{1,3,4,5}, Mod[#,8]]&]  (* Harvey P. Dale, Mar 09 2011 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 4, 5, 9}, 60] (* Bruno Berselli, Jul 17 2012 *)

makelist(2*n-1-(3+(-1)^n)*(1+%i^(n*(n+1)))/4,n,1,60);

Vec((1+2*x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

From Bruno Berselli, Jul 17 2012: (Start)
G.f.: x*(1+2*x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-1 -(3+(-1)^n)*(1+i^(n*(n+1)))/4, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047621(k), a(2k-1) = A047461(k). (End)
E.g.f.: (6 + sin(x) - 2*cos(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - (4+3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 24 2021

A091474 Decimal expansion of Pi^2 * (2-sqrt(2))/32.

Original entry on oeis.org

1, 8, 0, 6, 7, 1, 2, 6, 2, 5, 9, 0, 6, 5, 4, 9, 4, 2, 7, 9, 2, 3, 0, 8, 1, 2, 8, 9, 8, 1, 6, 7, 1, 6, 1, 5, 3, 3, 7, 1, 1, 4, 5, 7, 1, 0, 1, 8, 2, 9, 6, 7, 6, 6, 2, 6, 6, 2, 4, 0, 7, 9, 4, 2, 9, 3, 7, 5, 8, 5, 6, 6, 2, 2, 4, 1, 3, 3, 0, 0, 1, 7, 7, 0, 8, 9, 8, 2, 5, 4, 1, 5, 0, 4, 8, 3, 7, 9, 9, 7, 0, 7

Offset: 0

Eric W. Weisstein, Jan 13 2004

			0.18067126259065494279230812898167161533711457101829...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Definite Integral.
Index entries for transcendental numbers

Cf. A047621.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)^2*(2-Sqrt(2))/32; // G. C. Greubel, Oct 01 2018

RealDigits[Pi^2 (2-Sqrt[2])/32,10,120][[1]] (* Harvey P. Dale, Nov 18 2013 *)

default(realprecision, 100); Pi^2*(2-sqrt(2))/32 \\ G. C. Greubel, Oct 01 2018

Equals Integral_{t=0..1} t^2 * log(t)/((t^2 - 1)*(t^4 + 1)) dt.

Equals Sum_{k>=1} 1/A047621(k)^2. - Amiram Eldar, Apr 17 2022

A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.

Original entry on oeis.org

2, 3, 0, 1, 4, 5, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 18, 19, 16, 17, 20, 21, 22, 23, 26, 27, 24, 25, 28, 29, 30, 31, 34, 35, 32, 33, 36, 37, 38, 39, 42, 43, 40, 41, 44, 45, 46, 47, 50, 51, 48, 49, 52, 53, 54, 55, 58, 59, 56, 57, 60, 61, 62, 63, 66, 67, 64

Offset: 0

Franck Maminirina Ramaharo, Aug 01 2018

Write n in base 8, then apply the following substitution to the rightmost digit: '0'->'2, '1'->'3', and vice versa. Convert back to decimal.
A self-inverse permutation: a(a(n)) = n.
Array whose columns are, in this order, A047463, A047621, A047451 and A047522, read by rows.

			a(25) = a('3'1') = '3'3' = 27.
a(26) = a('3'2') = '3'0' = 24.
a(27) = a('3'3') = '3'1' = 25.
a(28) = a('3'4') = '3'4' = 28.
a(29) = a('3'5') = '3'5' = 29.
The sequence as array read by rows:
  A047463, A047621, A047451, A047522;
        2,       3,       0,       1;
        4,       5,       6,       7;
       10,      11,       8,       9;
       12,      13,      14,      15;
       18,      19,      16,      17;
       20,      21,      22,      23;
       26,      27,      24,      25;
       28,      29,      30,      31;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
OEIS wiki, Lodumo transform.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A047225, A047243, A047257, A064429, A080412, A159959, A026185.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1)))); // G. C. Greubel, Sep 25 2018

Table[(4*(Floor[1/4 Mod[2*n + 4, 8]] - Floor[1/4 Mod[n + 2, 8]]) + 2*n)/2, {n, 0, 100}]
f[n_] := Block[{id = IntegerDigits[n, 8]}, FromDigits[ Join[Most@ id /. {{} -> {0}}, {id[[-1]] /. {0 -> 2, 1 -> 3, 2 -> 0, 3 -> 1}}], 8]]; Array[f, 67, 0] (* or *)
CoefficientList[ Series[(x^7 + x^5 + 3x^3 - 2x^2 - x + 2)/((x - 1)^2 (x^6 + x^4 + x^2 + 1)), {x, 0, 70}], x] (* or *)
LinearRecurrence[{2, -2, 2, -2, 2, -2, 2, -1}, {2, 3, 0, 1, 4, 5, 6, 7}, 70] (* Robert G. Wilson v, Aug 01 2018 *)

makelist((4*(floor(mod(2*n + 4, 8)/4) - floor(mod(n + 2, 8)/4)) + 2*n)/2, n, 0, 100);

my(x='x+O('x^100)); Vec((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1))) \\ G. C. Greubel, Sep 25 2018

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - a(n-8), n > 7.
a(n) = (4*(floor(((2*n + 4) mod 8)/4) - floor(((n + 2) mod 8)/4)) + 2*n)/2.
a(n) = lod_4(A047247(n+1)).
a(4*n) = A047463(n+1).
a(4*n+1) = A047621(n+1).
a(4*n+2) = A047451(n+1).
a(4*n+3) = A047522(n+1).
a(A042948(n)) = A047596(n+1).
a(A042964(n+1)) = A047551(n+1).
G.f.: (x^7 + x^5 + 3*x^3 - 2*x^2 - x + 2)/((x-1)^2 * (x^2+1) * (x^4+1)).
E.g.f.: x*exp(x) + cos(x) + sin(x) + cos(x/sqrt(2))*cosh(x/sqrt(2)) + (sqrt(2)*cos(x/sqrt(2)) - sin(x/sqrt(2)))*sinh(x/sqrt(2)).
a(n+8) = a(n) + 8 . - Philippe Deléham, Mar 09 2023
Sum_{n>=3} (-1)^(n+1)/a(n) = 1/6 + log(2). - Amiram Eldar, Mar 12 2023

A203575 Array of certain four complete residue classes (nonnegative members), read by SW-NE antidiagonals.

Original entry on oeis.org

0, 1, 4, 2, 7, 8, 3, 6, 9, 12, 5, 10, 15, 16, 11, 14, 17, 20, 13, 18, 23, 24, 19, 22, 25, 28, 21, 26, 31, 32, 27, 30, 33, 36, 29, 34, 39, 40, 35, 38, 41, 44, 37, 42, 47, 48, 43, 46

Offset: 1

Wolfdieter Lang, Jan 12 2012

See A193682 for the sequence called P_4, with period length 8, which defines the four complete residue classes [m], m = 0,1,2,3, via the equivalence relation  p==q iff P_4(p) = P_4(q).
See a comment on A203571 for the general P_k sequences, and the multiplicative (but not additive) structure of these residue classes.
The row length sequence of this tabf array is [1,2,3,4,4,4,...].
This array defines a certain permutation of the nonnegative integers.

			The array starts
n\m  1   2   3   4
1:   0
2:   1   4
3:   2   7   8
4:   3   6   9  12
5:   5  10  15  16
6:  11  14  17  20
7:  13  18  23  24
8:  19  22  25  28
9:  21  26  31  32
10: 27  30  33  36
...
The sequence P_4(n)=A193682(n), n>=0, is repeated 0, 1, 2, 3, 0, 3, 2, 1, with period length 8. P_4(6)=2, hence 6 belongs to class [2].
Multiplicative structure: 11*23 == 3*1 = 3. Indeed: P_4(11*23) = P_4(253) = P_(5), because 253==5(mod 8), and P_(5)= 3, hence 11*23 belongs to class 3. In general, P_4(p*q) = P_4(P_4(p)*P_4(q)).

Cf.A193682, A088520 (k=3), A090298 (k=5), A092260 (k=6), A113807 (k=7).

The nonnegative members of the four complete residue classes are (see a comment above for their definition):
[0]: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36,...  (A008586)
[1]: 1, 7, 9, 15, 17, 23, 25, 31, 33, 39,...  (A047522)
[2]: 2, 6, 10, 14, 18, 22, 26, 30, 34, 38,... (A016825)
[3]: 3, 5, 11, 13, 19, 21, 27, 29, 35, 37,... (A047621)
In each class the corresponding negative numbers should be included.

cp's OEIS Frontend

A306220 a(n) is the smallest prime p such that Kronecker(-n,p) = -1.

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A047596 Numbers that are congruent to {2, 3, 4, 5} mod 8.

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A047599 Numbers that are congruent to {0, 3, 4, 5} mod 8.

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A329095 Odd numbers k such that x^2 == 2 (mod k) has no solution.

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A047499 Numbers that are congruent to {3, 4, 5, 7} mod 8.

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A047600 Numbers that are congruent to {1, 3, 4, 5} mod 8.

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A091474 Decimal expansion of Pi^2 * (2-sqrt(2))/32.

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