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.

A066958 Number of ordered primitive 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, 0, 3, 6, 6, 3, 9, 0, 9, 6, 6, 0, 15, 6, 3, 12, 9, 0, 15, 0, 15, 9, 6, 6, 18, 6, 3, 12, 18, 6, 21, 0, 9, 12, 6, 6, 27, 6, 9, 12, 18, 6, 15, 12, 18, 18, 6, 0, 33, 0, 15, 18, 18, 9, 18, 12, 9, 18, 18, 0, 39, 6, 9, 24, 18, 12, 18, 0, 27, 18, 15, 6, 33, 12, 6, 24, 30
Offset: 1

Views

Author

Colin Mallows, Jan 26 2002

Keywords

Links

Crossrefs

Cf. A066851.

Extensions

More terms from Vladeta Jovovic, Apr 14 2002

A374969 Number of ordered 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, 23, 0, 36, 0, 42, 24, 20, 0, 80, 16, 24, 32, 62, 0, 104, 0, 72, 40, 32, 48, 151, 0, 36, 48, 148, 0, 152, 0, 102, 120, 44, 0, 242, 36, 120, 64, 122, 0, 200, 80, 216, 72, 56, 0, 396, 0, 60, 176, 201, 96, 248, 0, 162, 88, 280, 0, 486, 0, 72, 208, 182
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

Examples

			a(6) = 4 since there are solutions (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2).

Links

Crossrefs

Cf. A000005, A000203, A066851, A374970.

Programs

  • PARI
    a(n) = sum(x=1, n, sum(y=1, n, sum(z=1, n, sum(w=1, n, x*y+y*z+z*w+w*x==n))));
  • Python
    from math import prod
from sympy import factorint
def A374969(n):
    f = factorint(n).items()
    return (n+1)*prod(e+1 for p,e in f)-(prod((p**(e+1)-1)//(p-1) for p,e in f)<<1) # Chai Wah Wu, Jul 26 2024

Formula

a(n) = (n+1)*A000005(n)-2*A000203(n). - Chai Wah Wu, Jul 26 2024

A375003 Number of ordered 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, 5, 12, 0, 16, 12, 24, 8, 0, 12, 34, 0, 24, 12, 30, 12, 16, 0, 36, 32, 24, 12, 32, 6, 36, 20, 36, 12, 40, 18, 72, 9, 0, 24, 64, 24, 48, 36, 30, 24, 56, 12, 72, 8, 48, 36, 70, 24, 60, 40, 54, 24, 40, 12, 120, 62, 24, 24, 80, 24, 96
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2024

Keywords

Examples

			a(9) = 4 since there are solutions (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2).

Links

Crossrefs

Cf. A066851, A374969, A375004.

Programs

  • PARI
    a(n) = sum(x=1, n, sum(y=1, n, sum(z=1, n, sum(w=1, n, x*y+x*z+x*w+y*z+y*w+z*w==n))));
  • Python
    from sympy import divisors, integer_nthroot
def A375003(n):
    k = 0
    for c in range(1,n-1):
        for d in divisors(c,generator=True):
            for x in range(1,d):
                xy = x*(d-x)
                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)//2
                        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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