A000161 -id:A000161 - cp's OEIS frontend

A124983 Nonprime numbers == 1 (mod 4) with a unique partition as a sum of 2 squares x^2 + y^2.

1, 9, 45, 49, 81, 117, 121, 153, 245, 261, 333, 361, 369, 405, 441, 477, 529, 549, 605, 637, 657, 729, 801, 833, 873, 909, 961, 981, 1017, 1053, 1089, 1233, 1341, 1377, 1413, 1421, 1557, 1573, 1629, 1737, 1773, 1805, 1813, 1849, 2009, 2057, 2061, 2097, 2169

Offset: 1

Artur Jasinski, Nov 16 2006

nonn

Intersection of A124982 and A125018. - Michel Marcus, Nov 02 2013

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

Cf. A000161, A002145, A124982, A125018, A125020.

Select[4 * Range[0, 500] + 1, !PrimeQ[#] && Length @ PowersRepresentations[#, 2, 2] == 1 &] (* Amiram Eldar, Mar 12 2020 *)

isok(n)= {if (isprime(n) || (n % 4 != 1), return (0)); A000161(n) == 1;} \\ Michel Marcus, Nov 02 2013

More terms from Michel Marcus, Nov 02 2013

A216284 Number of solutions to the equation x^4+y^4 = n with x >= y > 0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

V. Raman, Sep 03 2012

nonn

			From _Antti Karttunen_, Aug 28 2017: (Start)
For n = 2 there is one solution: 2 = 1^4 + 1^4, thus a(2) = 1.
For n = 17 there is one solution: 17 = 2^4 + 1^4, thus a(17) = 1.
For n = 635318657 we have two solutions: 635318657 = 158^4 + 59^4 = 134^4 + 133^4, thus a(635318657) = 2. Note that this is the first point where the sequence attains value greater than 1. See _Charles R Greathouse IV_'s Jan 12 2017 comment in A216280.
(End)

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

Cf. A025455, A025446, A000161, A025426, A216280.

(define (A216284 n) (let loop ((x (A255270 n)) (s 0)) (let* ((x4 (A000583 x)) (y4 (- n x4))) (if (< x4 y4) s (loop (- x 1) (+ s (if (and (> y4 0) (= (A000583 (A255270 y4)) y4)) 1 0))))))) ;; Antti Karttunen, Aug 28 2017

a(n) <= A216280(n). - Antti Karttunen, Aug 28 2017

Definition edited to match the given data and the second part of offset (635318657) explicitly added by Antti Karttunen, Aug 28 2017

A259285 Expansion of psi(x^2) * f(x, x^7) in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 2, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 3, 0, 1, 0, 1, 3, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 1, 0

Offset: 0

Michael Somos, Jun 23 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + x^2 + x^3 + x^6 + 2*x^7 + x^9 + x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^13 + q^29 + q^45 + q^61 + q^109 + 2*q^125 + q^157 + q^173 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000161, A025426, A025435, A025441, A134343, A259287.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^8] QPochhammer[ -x^2, x^8] QPochhammer[ -x^6, x^8] QPochhammer[ -x^7, x^8] QPochhammer[x^8]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 + x^(8 k - 1)) (1 + x^(8 k - 2)) (1 + x^(8 k - 6)) (1 + x^(8 k - 7)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];

{a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 13); for(u = (s+3)\-8, (s-3)\8, if( issquare( m - (8*u + 3)^2, &x) && (x%8==2 || x%8==6), c++))); c};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -1, 0, 0, 1, 0, -1, -1, 2, -1, -1, 0, 1, 0, 0, -1][k%16 + 1], 1 + x * O(x^n)), n))};

Number of solutions to 16*n + 13 = (8*u + 3)^2 + (8*v + 2)^2 where u,v in Z.
Euler transform of period 16 sequence [ 1, 0, 0, -1, 0, 1, 1, -2, 1, 1, 0, -1, 0, 0, 1, -2, ...].
a(9*n + 2) = A259287(n). a(9*n + 5) = a(9*n + 8) = 0.
-2 * a(n) = A134343(4*n + 3). a(n) = A000161(16*n + 13) = A025426(16*n + 13) = A025435(16*n + 13) = A025441(16*n + 13).

A259287 Expansion of psi(x^2) * f(x^3, x^5) in powers of x where psi(), f(, ) are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 3, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 0, 1, 2, 0, 1, 1, 0, 1, 3, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 2, 0, 2, 2, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1

Offset: 0

Michael Somos, Jun 23 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x^2 + x^3 + 2*x^5 + x^6 + x^7 + x^9 + x^11 + x^12 + x^14 + ...
G.f. = q^5 + q^37 + q^53 + 2*q^85 + q^101 + q^117 + q^149 + q^181 + q^197 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000161, A025426, A025435, A025441, A134343, A259285.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^8] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ -x^6, x^8] QPochhammer[x^8]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[(1 + x^(8 k - 2)) (1 + x^(8 k - 3)) (1 + x^(8 k - 5)) (1 + x^(8 k - 6)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];

{a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 5); for(u = (s+1)\-8, (s-1)\8, if( issquare( m - (8*u + 1)^2, &x) && (x%8==2 || x%8==6), c++))); c};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, 0, -1, -1, 1, -1, 0, 0, 2, 0, 0, -1, 1, -1, -1, 0][k%16 + 1], 1 + x * O(x^n)), n))};

Number of solutions to 16*n + 5 = (8*u + 1)^2 + (8*v + 2)^2 where u,v in Z.
Euler transform of period 16 sequence [ 0, 1, 1, -1, 1, 0, 0, -2, 0, 0, 1, -1, 1, 1, 0, -2, ...].
a(9*n + 1) = a(9*n + 4) = 0. a(9*n + 7) = A259285(n).
-2 * a(n) = A134343(4*n + 1). a(n) = A000161(16*n + 5) = A025426(16*n + 5) = A025435(16*n + 5) = A025441(16*n + 5).

A328803 The minimum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 4, 5, 4, 5, 6, 6, 5, 6, 7, 8, 8, 6, 7, 8, 9, 9, 7, 8, 10, 9, 10, 11, 8, 9, 10, 12, 11, 12, 12, 9, 10, 11, 13, 12, 13, 14, 10, 11, 12, 14, 13, 15, 14, 15, 11, 12, 13, 16, 14, 16, 15, 12, 13, 16, 14, 17, 15, 17, 16, 18, 18, 13, 14, 15, 16

Offset: 1

Peter Kagey, Oct 27 2019

			For n = 14, A001481(14) = 25 = 0^2 + 5^2 = 3^2 + 4^2, so a(14) = min{0+5, 3+4} = 5.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000161, A001481.

N:= 1000: # for terms where A001481(n)<=N
for s from 0 to isqrt(N) do
   for i from 0 to s/2 do
      t:= i^2 + (s-i)^2;
      if t > N then break fi;
      if not assigned(R[t]) then R[t]:= s fi;
od od:
A1481:= sort(map(op, [indices(R)])):
seq(R[i],i=A1481); # Robert Israel, Oct 28 2019

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
from sympy import factorint
def A328803_gen(): # generator of terms
    return map(lambda n: min((a+b for a, b in diop_DN(-1,n))), filter(lambda n:(lambda m:all(d&3!=3 or m[d]&1==0 for d in m))(factorint(n)), count(0)))
A328803_list = list(islice(A328803_gen(),30)) # Chai Wah Wu, Sep 09 2022

A063665 Number of ways 1/n can be written as 1/x^2 + 1/y^2 with y >= x >= 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Jul 25 2001

nonn

Number of ordered pairs (x,y), with n = (x^2)(y^2)/(x^2 + y^2) and y >= x > 0. - Antti Karttunen, Nov 07 2018

			a(90)=1 since 1/90 = 1/10^2 + 1/30^2
a(98)=2 since 1/98 = 1/10^2 + 1/70^2 = 1/14^2 + 1/14^2.
a(14400) = 3 since 1/14400 = 1/130^2 + 1/312^2 = 1/136^2 + 1/255^2 = 1/150^2 + 1/200^2. - _Antti Karttunen_, Nov 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000161, A025426, A018892, A063664.

A063665(n) = { my(s=0); for(x=1,n,for(y=x,n,if((n*(x*x+y*y)) == (x*x*y*y), s++))); (s); }; \\ Antti Karttunen, Nov 07 2018

A063665(n) = { my(s=0,y); for(x=sqrtint(n),n,my(x2=x*x); if((x2>n)&&issquare((n*x2)/(x2-n),&y)&&(1==denominator(y))&&(y>=x),s++)); (s); }; \\ Antti Karttunen, Nov 07 2018

Definition clarified by Antti Karttunen, Nov 07 2018

A102548 Number of positive integers <= n that are expressible in the form u^2+v^2, with u and v integers.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 19, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 24, 24, 25, 26, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29, 30, 30, 30, 31, 31, 31, 31, 32, 33, 34, 34

Offset: 1

Salvador Perez Gomez (pies314(AT)hotmail.com), Feb 24 2005

			a(8) = 5 because 1 = 0^2 + 1^2, 2 = 1^2 + 1^2, 4 = 0^2 + 2^2, 5 = 1^2 + 2^2, 8 = 2^2 + 2^2, but 3,6 and 7 are not of the form u^2 + v^2, with u and v integers.

David A. Corneth, Table of n, a(n) for n = 1..10000
Yoichi Motohashi, On the distribution of prime numbers which are of the form x^2+y^2+1, Acta Arith. 16 (1969/70), 351-363. MR0288086 (44 #5284). See Eq. (1.2).
D. Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant.

Cf. A000161, A000691, A001481, A229062.

a := proc(n) local aux,i,m,u,v; aux:=0; for i from 1 to n do m:=floor(sqrt(i/2)); for u from 0 to m do v:=sqrt(i-u^2); if (v = floor(v)) then aux:=aux+1; u:=m; end if; end do; end do; aux; end proc:

a[1]=1; a[n_]:= a[n]= a[n-1] + If[SquaresR[2, n]>0, 1, 0]; Table[a[n], {n,75}] (* Jean-François Alcover, Mar 31 2015 *)

first(n)= my(v = vector(n + 1), res = vector(n)); res[1] = 1; for(i = 0, sqrtint(n), for(j = i, sqrtint(n - i^2), v[i^2+j^2+1] = 1 ) ); for(i = 2, #res, res[i] = res[i-1] + v[i+1]; ); res \\ David A. Corneth, Jun 05 2020

from itertools import count, accumulate, islice
from sympy import factorint
def A102548_gen(): # generator of terms
    return accumulate(int(all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items())) for n in count(1))
A102548_list = list(islice(A102548_gen(),30)) # Chai Wah Wu, Jun 28 2022

From David A. Corneth, Jun 05 2020: (Start)
A000161(a(n)) > 0.
a(n) = (partial sum of A229062 up to n) - 1. (End)
a(n) = n/sqrt(log n) * (K + B2/log n + O(1/log^2 n)), where K = A064533 and B2 = A227158. In particular, a(n) ~ Kn/sqrt(log n). - Charles R Greathouse IV, Dec 03 2022

Name clarified by David A. Corneth, Jun 05 2020

A124134 Positive integers n such that Fibonacci(n) = a^2 + b^2, where a, b are integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122, 123, 125, 127

Offset: 1

Melvin J. Knight (melknightdr(AT)verizon.net), Nov 30 2006

nonn

All odd numbers are in this sequence, since the Fibonacci number with index 2m+1 is the sum of the squares of the two Fibonacci numbers with indices m and m+1. Those with even indices ultimately depend on certain Lucas numbers being the sum of two squares (see A124132). Joint work with Kevin O'Bryant and Dennis Eichhorn.

Numbers n such that Fibonacci(n) or Fibonacci(n)/2 is a square are only 0, 1, 2, 3, 6, 12. So a and b must be distinct and nonzero for all values of this sequence except 1, 2, 3, 6, 12. - Altug Alkan, May 04 2016

			14 is in the sequence because F_14=377=11^2+16^2.
16 is not in the sequence because F_16=987 is congruent to 3 (mod 4).

Chai Wah Wu, Table of n, a(n) for n = 1..1322 (terms 1..210 from Joerg Arndt)

Cf. A124132, A000045, A000161, A001481.

a124134 n = a124134_list !! (n-1)
a124134_list = filter ((> 0) . a000161 . a000045) [1..]
-- Reinhard Zumkeller, Oct 10 2013

Select[Range@ 128, SquaresR[2, Fibonacci@ #] > 0 &] (* Michael De Vlieger, May 04 2016 *)

for(n=1, 10^6, t=fibonacci(n); s=sqrtint(t); forstep(i=s, 1, -1, if(issquare(t-i*i), print1(n, ", "); break))) \\ Ralf Stephan, Sep 15 2013

is2s(n)={my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1; } \\ see A001481
for(n=1, 10^6, if(is2s(fibonacci(n)), print1(n, ", "))); \\ Joerg Arndt, Sep 15 2013

from itertools import count, islice
from sympy import factorint, fibonacci
def A124134_gen(): # generator of terms
    return filter(lambda n:n & 1 or all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(fibonacci(n)).items()),count(1))
A124134_list = list(islice(A124134_gen(),30)) # Chai Wah Wu, Jun 27 2022

Intersection of A000045 and A001481.

A000161(A000045(a(n))) > 0. - Reinhard Zumkeller, Oct 10 2013

More terms from Ralf Stephan, Sep 15 2013

A133102 Number of partitions of n^3 into n distinct nonzero squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 5, 20, 56, 112, 268, 618, 1922, 8531, 29021, 100407, 321531, 899618, 2937312, 9295401, 31615059, 117365818, 403433963, 1417579281, 4848439367, 15960316056, 55180971700, 190251417034, 670818005444, 2429973932322

Offset: 1

Hugo Pfoertner, Sep 12 2007

nonn

			a(6) = 3 because there are 3 ways to express 6^3 = 216 as a sum of 6 distinct nonzero squares: 216 = 1^2 + 2^2 + 4^2 + 5^2 + 7^2 + 11^2 = 1^2 + 3^2 + 5^2 + 6^2 + 8^2 + 9^2 = 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 9^2.

Robert Gerbicz, May 09 2008, Table of n, a(n) for n = 1..40

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.

Cf. A133103 (number of ways to express n^3 as a sum of n nonzero squares), A133105 (number of ways to express n^4 as a sum of n distinct nonzero squares).

a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

More terms from Robert Gerbicz, May 09 2008

A133103 Number of partitions of n^3 into n nonzero squares.

Original entry on oeis.org

1, 1, 2, 1, 10, 34, 156, 734, 3599, 18956, 99893, 548373, 3078558, 17510598, 101960454, 599522778, 3565904170, 21438347021, 129905092421, 794292345434, 4890875249113, 30326545789640, 189195772457341, 1187032920371427

Offset: 1

Hugo Pfoertner, Sep 11 2007

nonn

			a(2)=1 because the only way to express 2^3 = 8 as a sum of two squares is 8 = 2^2 + 2^2.
a(3)=2 because 3^3 = 27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2.

Robert Gerbicz, May 09 2008, Table of n, a(n) for n = 1..40

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.

Cf. A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares), A133104 (number of ways to express n^4 as a sum of n nonzero squares).

a(i, n, k)=local(s, j); if(k==1, if(issquare(n), return(1), return(0)), s=0; for(j=ceil(sqrt(n/k)), min(i, floor(sqrt(n-k+1))), s+=a(j, n-j^2, k-1)); return(s)) for(n=1,50, m=n^3; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

More terms from Robert Gerbicz, May 09 2008

cp's OEIS Frontend

A124983 Nonprime numbers == 1 (mod 4) with a unique partition as a sum of 2 squares x^2 + y^2.

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A216284 Number of solutions to the equation x^4+y^4 = n with x >= y > 0.

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A259285 Expansion of psi(x^2) * f(x, x^7) in powers of x where psi(), f(,) are Ramanujan theta functions.

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A259287 Expansion of psi(x^2) * f(x^3, x^5) in powers of x where psi(), f(, ) are Ramanujan theta functions.

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A328803 The minimum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

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A063665 Number of ways 1/n can be written as 1/x^2 + 1/y^2 with y >= x >= 1.

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A102548 Number of positive integers <= n that are expressible in the form u^2+v^2, with u and v integers.

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