cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A066851 Number of ordered solutions (x,y,z) to xy + yz + zx = n with x,y,z >= 1.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 3, 3, 3, 0, 9, 1, 3, 6, 6, 3, 9, 0, 9, 9, 6, 0, 15, 6, 3, 12, 10, 3, 15, 0, 15, 12, 6, 6, 18, 9, 3, 12, 18, 6, 21, 0, 9, 21, 9, 6, 27, 7, 9, 12, 18, 9, 15, 12, 18, 24, 6, 0, 33, 6, 15, 18, 21, 12, 18, 12, 9, 27, 18, 0, 39, 9, 9, 24, 19, 21, 18, 0, 27, 27, 18, 6, 33, 18, 6
Offset: 1

Views

Author

Colin Mallows, Jan 24 2002

Keywords

Examples

			a(5) = 3 since there are solutions (2,1,1), (1,2,1), (1,1,2).

Links

Crossrefs

a(A025052(n))=0. Cf. A066958.

Extensions

More terms from Vladeta Jovovic, Jan 25 2002

A374970 Number of ordered primitive solutions (x,y,z,w) to x*y + y*z + z*w + w*x = n with x,y,z,w >= 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 0, 6, 4, 8, 0, 22, 0, 12, 16, 22, 0, 36, 0, 42, 24, 20, 0, 76, 16, 24, 32, 62, 0, 104, 0, 66, 40, 32, 48, 146, 0, 36, 48, 140, 0, 152, 0, 102, 120, 44, 0, 220, 36, 120, 64, 122, 0, 196, 80, 204, 72, 56, 0, 380, 0, 60, 176, 178, 96, 248, 0, 162, 88, 280, 0, 444, 0, 72, 208, 182, 120, 296, 0, 396
Offset: 1

Views

Author

Seiichi Manyama, Jul 26 2024

Keywords

Comments

a(n) = 0 if and only if n = 1 or n is prime. - Chai Wah Wu, Jul 26 2024

Crossrefs

Cf. A066958, A374969.

Programs

  • PARI
    a(n) = sum(x=1, n, sum(y=1, n, sum(z=1, n, sum(w=1, n, (gcd([x, y, z, w])==1)*(x*y+y*z+z*w+w*x==n)))));
  • Python
    from math import gcd
from sympy import divisors
def A374970(n): return sum(1 for d in divisors(n,generator=True) for x in range(1,d) for y in range(1,n//d) if gcd(x,y,d-x,n//d-y)==1) # Chai Wah Wu, Jul 26 2024

A375004 Number of ordered primitive solutions (x,y,z,w) to x*y + x*z + x*w + y*z + y*w + z*w = n with x,y,z,w >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 4, 6, 0, 4, 0, 12, 8, 0, 0, 16, 6, 12, 4, 12, 0, 16, 12, 24, 8, 0, 12, 34, 0, 24, 8, 30, 12, 16, 0, 36, 32, 24, 12, 32, 6, 36, 16, 36, 12, 40, 12, 72, 8, 0, 24, 64, 24, 48, 32, 30, 24, 56, 12, 72, 8, 48, 24, 70, 24, 60, 32, 54, 24, 40, 12, 120, 62, 24, 24, 76, 24, 96, 32
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2024

Keywords

Links

Crossrefs

Cf. A066958, A374970, A375003.

Programs

  • PARI
    a(n) = sum(x=1, n, sum(y=1, n, sum(z=1, n, sum(w=1, n, (gcd([x, y, z, w])==1)*(x*y+x*z+x*w+y*z+y*w+z*w==n)))));
  • Python
    from math import gcd
from sympy import divisors, integer_nthroot
def A375004(n):
    k = 0
    for c in range(1,n-1):
        for d in divisors(c,generator=True):
            for x in range(1,d):
                y = d-x
                xy = x*y
                a = (c//d)**2
                b = a-(n-c-xy<<2)
                if b>=0:
                    q,r = integer_nthroot(b,2)
                    if r:
                        w = c//d+q>>1
                        z = c//d-w
                        if 1<=w>1
                            z = c//d-w
                            if 1<=wChai Wah Wu, Jul 27 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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