A008784 -id:A008784 - cp's OEIS frontend

A037942 Numbers of the form x^2 + y^2 with x >= 0, y >= 0, gcd(x,y)=1, with multiplicity.

1, 2, 5, 10, 13, 17, 25, 26, 29, 34, 37, 41, 50, 53, 58, 61, 65, 65, 73, 74, 82, 85, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 130, 137, 145, 145, 146, 149, 157, 169, 170, 170, 173, 178, 181, 185, 185, 193, 194

Offset: 1

Nissim Broudo (broudo(AT)brain.math.fau.edu)

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Eric Weisstein's World of Mathematics, Postage Stamp Problem.

If repetitions are deleted, same as A008784.

Cf. A224770 (multiplicity 2), A281877 (multiplicity >= 3).

s=14; v=[]; for(y=0,s,for(x=0,y,if(gcd(x,y)==1,r=x*x+y*y; if(r<=s*s,v=concat(v,r))))); vecsort(v) \\ Craig Clapp (craig_clapp(AT)earthlink.net), Apr 30 2010

Missing second instance of 85 (6^2+7^2 = 2^2+9^2 = 85) added by Craig Clapp (craig_clapp(AT)earthlink.net), Apr 30 2010

A103215 Numbers congruent to {1, 2, 5, 10, 13, 17} mod 24.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 25, 26, 29, 34, 37, 41, 49, 50, 53, 58, 61, 65, 73, 74, 77, 82, 85, 89, 97, 98, 101, 106, 109, 113, 121, 122, 125, 130, 133, 137, 145, 146, 149, 154, 157, 161, 169, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 209, 217, 218, 221, 226

Offset: 1

Ralf Stephan, Jan 28 2005

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

Union of A008784 and A103216.

Cf. A054895.

a103215 n = a103215_list !! (n-1)
a103215_list = [1,2,5,10,13,17] ++ map (+ 24) a103215_list
-- Reinhard Zumkeller, Jul 05 2013

[n : n in [0..300] | n mod 24 in [1, 2, 5, 10, 13, 17]]; // Wesley Ivan Hurt, Jul 22 2016

A103215:=n->24*floor(n/6)+[1, 2, 5, 10, 13, 17][(n mod 6)+1]: seq(A103215(n), n=0..100); # Wesley Ivan Hurt, Jul 22 2016

Select[Range[300], MemberQ[{1,2,5,10,13,17}, Mod[#,24]]&] (* or *) LinearRecurrence[{1,0,0,0,0,1,-1}, {1,2,5,10,13,17,25}, 60] (* Harvey P. Dale, Feb 19 2015 *)

G.f.: x*(1+x+3*x^2+5*x^3+3*x^4+4*x^5+7*x^6) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Jul 02 2011
a(1)=1, a(2)=2, a(3)=5, a(4)=10, a(5)=13, a(6)=17, a(7)=25, a(n) = a(n-1)+ a(n-6)-a(n-7) for n>7. - Harvey P. Dale, Feb 19 2015
From Wesley Ivan Hurt, Jul 22 2016: (Start)
a(n) = a(n-6) + 24 for n>6.
a(n) = (12*n - 18 + cos(n*Pi/3) - 3*cos(2*n*Pi/3) - cos(n*Pi) + 2*sqrt(3)*sin(n*Pi/3) + 2*sqrt(3)*sin(2*n*Pi/3))/3.
a(6k) = 24k-7, a(6k-1) = 24k-11, a(6k-2) = 24k-14, a(6k-3) = 24k-19, a(6k-4) = 24k-22, a(6k-5) = 24k-23. (End)

A166134 a(n+1) is the smallest divisor of a(n)^2+1 that does not yet appear in the sequence, with a(1) = 1.

Original entry on oeis.org

1, 2, 5, 13, 10, 101, 5101, 26, 677, 45833, 65, 2113, 446477, 130, 16901, 41, 29, 421, 17, 58, 673, 45293, 25, 313, 97, 941, 34057, 50, 61, 1861, 1229, 773, 59753, 89, 34, 1157, 82, 269, 194, 617, 38069, 55740337, 145, 10513, 11052317, 12215371106849

Offset: 1

Franklin T. Adams-Watters, Oct 07 2009

nonn

All members of the sequence can be represented as the sum of two relatively prime numbers (A008784). It appears that the sequence is infinite and that all such numbers are present.

			After a(4)=13, the divisors of 13^2+1=170 are 1,2, 5, 10, 17, 34, 85, 170. 1, 2, and 5 have already occurred, so a(5) = 10.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A166133, A008784, A031439, A002522.

Nest[Append[#, Min[Complement[Divisors[#[[-1]]^2 + 1], #]]] &, {1}, 45] (* Ivan Neretin, Sep 03 2015 *)

invec(v,x,n)=for(i=1,n,if(v[i]==x,return(1)));0
bl(n)={local(v,d,ds);
v=vector(n,i,1);
for(i=2,n,
ds=divisors(v[i-1]^2+1);
for(k=2,#ds,d=ds[k];if(!invec(v,d,i-1),v[i]=d;break)));
v}

A225771 Numbers that are positive integer divisors of 1 + 2*x^2 where x is a positive integer.

Original entry on oeis.org

1, 3, 9, 11, 17, 19, 27, 33, 41, 43, 51, 57, 59, 67, 73, 81, 83, 89, 97, 99, 107, 113, 121, 123, 129, 131, 137, 139, 153, 163, 171, 177, 179, 187, 193, 201, 209, 211, 219, 227, 233, 241, 243, 249, 251, 257, 267, 281, 283, 289, 291, 297, 307, 313, 321, 323

Offset: 1

Michael Somos, Jul 26 2013

nonn

This sequence is case k=2, A008784 is case k=1, A004613 is case k=4 of divisors of 1 + k*x^2.
From Peter M. Chema, May 08 2017 (Start): Also gives the body diagonals of all primitive Pythagorean quadruples that define square prisms, with sides [b, b, and c] and diagonal d, such that 2*b^2 + c^2 = d^2. E.g., sides [2, 2, 1], diagonal 3 = a(2); [4, 4, 7], 9 = a(3); [6, 6, 7], 11 = a(4); [12, 12, 1], 17 = a(5); [6, 6, 17] 19 = a(6); [10, 10, 23], 27 = a(7); [20, 20, 17], 33 = a(8); [24, 24, 23], 41  = a(9)... (a subsequence of A096910) (End)
Editorial note: The above comment would be better expressed by talking about right tetrahedra (also called trirectangular tetrahedra), that is, tetrahedra with vertices (b 0 0), (0 c 0), (0 0 d) (here b=c).  These are the correct generalizations of Pythagorean triangles. N. J. A. Sloane, May 08 2017
From Frank M Jackson, May 23 2017: (Start)
Starting at a(2)=3, this gives the shortest side of a primitive Heronian triangle whose perimeter is 4 times its shortest side. Aka a primitive integer Roberts triangle (see Buchholz link).
Also odd and primitive terms generated by x^2 + 2y^2 with x>0 and y>0.
Also integers with all prime divisors congruent to 1 or 3 (mod 8). (End)

Giovanni Resta, Table of n, a(n) for n = 1..1000
Ralph H. Buchholz, On Triangles with rational altitudes, angle bisectors or medians, Newcastle University (1989), 21-22.

Cf. A004613, A008784, A096910.

Select[Range[323], False =!= Reduce[1 + 2*x^2 == # y , {x, y}, Integers] &] (* Giovanni Resta, Jul 28 2013 *)
Select[Range[323], OddQ[#]&&Intersection[{5, 7}, Mod[Divisors[#], 8]]=={} &] (* Frank M Jackson, May 23 2017 *)

{isa(n) = if( n<2, n==1, for( k=1, n\2, if( (1 + 2*k^2)%n == 0, return(1))))} /* Michael Somos, Jul 28 2013 */

a(n) integers whose prime divisors are congruent to 1 or 3 (mod 8). - Carmine Suriano, Jan 09 2015; corrected by Frank M Jackson, May 23 2017

A034026 Numbers that are primitively or imprimitively represented by x^2+y^2, but not both.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153

Offset: 1

N. J. A. Sloane

nonn

Cf. A001481, A008784, A022544, A034023-.

A151926 Numbers n such that A151925(n) = 3.

Original entry on oeis.org

3, 6, 9, 11, 14, 18, 19, 21, 22, 27, 30, 33, 35, 38, 42, 43, 45, 46, 49, 51, 54, 57, 59, 62, 66, 67, 69, 70, 75, 77, 78, 81, 83, 86, 90, 91, 93, 94, 98, 99, 102, 105, 107, 110, 114, 115, 117, 118, 121, 123, 126, 129, 131, 133, 134, 138, 139, 141, 142, 147, 150, 153, 154, 155, 158

Offset: 1

N. J. A. Sloane, Aug 07 2009

nonn

N \ {A008784, numbers == 0,4,7 mod 8}.

A174051 Composite numbers of the form x^2+y^2, gcd(x,y) = 1.

Original entry on oeis.org

10, 25, 26, 34, 50, 58, 65, 74, 82, 85, 106, 122, 125, 130, 145, 146, 169, 170, 178, 185, 194, 202, 205, 218, 221, 226, 250, 265, 274, 289, 290, 298, 305, 314, 325, 338, 346, 362, 365, 370, 377, 386, 394, 410, 425, 442, 445, 458, 466, 481, 482, 485, 493, 505

Offset: 1

Michel Lagneau, Mar 06 2010

nonn

Composite numbers in A008784. - R. J. Mathar, Jul 08 2012

			10 is in the sequence because 10 = 1^2 + 3^2 = 2*5;
25 is in the sequence because 25 = 3^2 + 4^2 = 5*5;
65 is in the sequence because 65 = 1 + 8^2 = 4^2 + 7^2 = 5*13.

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

Cf. A002808, A002313, A037942.

with(numtheory):T:=array(0..50000000):U=array(0..50000000 ):k:=1:for x from 1 to 1000 do:for y from x to 1000 do:if type(x^2+y^2,prime)=false and gcd(x,y)=1 then T[k]:=x^2+y^2:k:=k+1:else fi: od :od:mini:=T[1]:ii:=1:for p from 1 to k-1 do:or n from 1 to k-1 do:if T[n] < mini then mini:= T[n]:ii:=n: indice:=U[n]: else fi:od:print(mini):T[ii]:= 99999999: ii:=1:mini:=T[1] :od:

list(lim)=my(v=List(), x2, t); lim\=1; for(x=3, sqrtint(lim-1), x2=x^2; for(y=1, min(x-1, sqrtint(lim-x2)), if(gcd(x, y)==1 && !isprime(t=x2+y^2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Jan 27 2018

A319813 a(n) is the smallest a such that n is divisible by a^n + 1, or 0 if no such a exists.

Original entry on oeis.org

1, 1, 2, 0, 4, 0, 6, 0, 2, 3, 10, 0, 12, 0, 14, 0, 16, 0, 18, 0, 5, 0, 22, 0, 4, 5, 2, 0, 28, 0, 30, 0, 32, 13, 34, 0, 36, 0, 17, 0, 40, 0, 42, 0, 14, 0, 46, 0, 6, 3, 50, 0, 52, 0, 19, 0, 8, 17, 58, 0, 60, 0, 5, 0, 64, 0, 66, 0, 68, 0, 70, 0, 72, 31, 14, 0, 76

Offset: 1

Jianing Song, Sep 28 2018

nonn

a(n) = 0 iff n is even and -1 is not a square modulo n, that is, n is even and not in A008784. For other n > 2, 2 <= a(n) <= n - 1.
a(p) = p - 1 for primes p. For composite n, a(n) = n - 1 iff gcd(n, phi(n)) = 1, that is, n is in A050384.
a(A006521(n)) = 2.

Jianing Song, Table of n, a(n) for n = 1..7000

Cf. A074792 (a^n - 1 instead of a^n + 1).

Cf. also A006521, A008784, A050384.

a(n) = if(!(n%2)&&!issquare(Mod(-1,n)), 0, my(i=1); while(Mod(i,n)^n!=n-1, i++); i)

For n = 9, 9 is divisible by a^9 + 1 implies a == 2 (mod 3), so a(9) = 2.
For n = 10, 10 is divisible by a^10 + 1 implies a == 3, 7 (mod 10), so a(10) = 3.
For n = 34, 34 is divisible by a^34 + 1 implies a == 13, 21 (mod 34), so a(34) = 13.

A319838 Numbers k such that A319100(k) is a power of 6.

Original entry on oeis.org

1, 2, 7, 9, 13, 14, 18, 19, 26, 27, 31, 37, 38, 43, 49, 54, 61, 62, 63, 67, 73, 74, 79, 81, 86, 91, 97, 98, 103, 109, 117, 122, 126, 127, 133, 134, 139, 146, 151, 157, 158, 162, 163, 169, 171, 181, 182, 189, 193, 194, 199, 206, 211, 217, 218, 223, 229, 234, 241

Offset: 1

Jianing Song, Sep 28 2018

nonn

Numbers k such that the number of solutions to x^6 == 1 (mod k) is a power of 6.
Also numbers k such that (Z/kZ)* has the same 2-rank and 3-rank, where (Z/kZ)* is the multiplicative group of integers modulo k, and the p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) with p being a prime number.
k is a term in this sequence iff v(2, k) = 0 or 1, v(3, k) = 0 or >= 2 and k is not divisible by any prime p == 5 (mod 6). Here v(p, k) is the p-adic valuation of k.
Sequence contains all primes p == 1 (mod 6) and their powers and all powers of 3 except for 3 itself.
Decompose the multiplicative group of integers modulo k as a product of cyclic groups C_{s_1} x C_{s_2} x ... x C_{s_m}, where s_i divides s_j for i < j, then k is a term iff s_1 is divisible by 6. For k = 1 or 2, (Z/kZ)* is the trivial group, s_1 does not exist so 1 and 2 are also terms.
If gcd(k_1, k_2) = 1 and both k_1 and k_2 are in this sequence, so is k_1*k_2. For example, 7 and 9 are both here so 7*9 = 63 is also here. Indeed, the number of solutions to x^6 == 1 (mod 7), x^6 == 1 (mod 9) and x^6 == 1 (mod 36) are 6, 6 and 36, respectively.
This is an analog of A008784, since k is a term there iff s_1 (defined as above) is divisible by 4 instead of 6. But on the other hand, if k is in A008784, so are all its divisors, while this is not true for this sequence. However, if k is here and k is not divisible by 9, then all its divisors are also here.
This is a also an analog of A192453 (s_1 divisible by 8).

			91 = 7*13 is a term since the number of solutions to x^6 == 1 (mod 91) is 36 = 6^2.
1197 = 7*9*19 is a term since the number of solutions to x^6 == 1 (mod 1197) is 216 = 6^3.

Jianing Song, Table of n, a(n) for n = 1..15917 (all terms below 100000)

Cf. A008784, A192453, A319100.

isA319838(n) = if(abs(n)==1||abs(n)==2, 1, my(i=znstar(n)[2]); !(i[#i]%6)==1, 1)

A337910 Integers of the form (the number of nonnegative bases m < n such that m^3 == m (mod n))/(the number of nonnegative bases m < n such that -m^3 == m (mod n)).

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 3, 5, 3, 1, 3, 9, 1, 3, 3, 5, 1, 3, 3, 3, 9, 3, 3, 15, 1, 1, 3, 9, 1, 3, 3, 5, 9, 1, 3, 9, 1, 3, 3, 5, 1, 9, 3, 9, 3, 3, 3, 15, 3, 1, 3, 3, 1, 3, 3, 15, 9, 1, 3, 9, 1, 3, 9, 5, 1, 9, 3, 3, 9, 3, 3, 15, 1, 1, 3, 9, 9, 3, 3, 5, 3, 1, 3, 27, 1, 3, 3, 15, 1, 3, 3, 9, 9, 3, 3, 15

Offset: 1

Juri-Stepan Gerasimov, Sep 29 2020

All members of a(n) are odd numbers. For n > 3, 1 <= a(n) < n.

Cf. A026741, A008784, A002313, A002145, A334006.

[#[m: m in [0..n-1] | m^3 mod n eq m]/#[m: m in [0..n-1] | -m^3 mod n eq m]: n in [1..96]];

cp's OEIS Frontend

A037942 Numbers of the form x^2 + y^2 with x >= 0, y >= 0, gcd(x,y)=1, with multiplicity.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A103215 Numbers congruent to {1, 2, 5, 10, 13, 17} mod 24.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

A166134 a(n+1) is the smallest divisor of a(n)^2+1 that does not yet appear in the sequence, with a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A225771 Numbers that are positive integer divisors of 1 + 2*x^2 where x is a positive integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A034026 Numbers that are primitively or imprimitively represented by x^2+y^2, but not both.

Views

Author

Keywords

Crossrefs

A151926 Numbers n such that A151925(n) = 3.

Views

Author

Keywords

Formula

A174051 Composite numbers of the form x^2+y^2, gcd(x,y) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A319813 a(n) is the smallest a such that n is divisible by a^n + 1, or 0 if no such a exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A319838 Numbers k such that A319100(k) is a power of 6.

Views

Author

Keywords

Comments