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 1-8 of 8 results.

A345691 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = n^4*s, where s is the population variance of the values of u^2+v^2.

Original entry on oeis.org

0, 0, 14, 48, 1266, 1988, 21514, 49392, 171032, 242964, 882708, 1487996, 3650020, 4913620, 9374594, 14382448, 29859148, 38410016, 71427550, 97525500, 147544988, 186821472, 320133640, 399015644, 605818854, 740618592, 1061345430, 1349418108, 2017326672, 2390222900
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor n^4 is to ensure that a(n) is an integer.
A345434(n) = n^2*mu where mu is the mean of the values of u^2+v^2.
s^(1/4) appears to grow linearly with n.

Crossrefs

Cf. A345434, A345687, A345688, A345689, A345690.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345691(n): return pvariance(n**2*(u**2+v**2) for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)))

A345692 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = u*x+v*y with u,v minimal; a(n) = m^2*s, where s is the population variance of the values of u and m is the number of such values.

Original entry on oeis.org

0, 2, 24, 68, 364, 504, 2040, 3606, 7664, 10422, 25764, 34226, 70836, 89994, 128532, 177276, 316844, 375952, 623024, 757604, 986742, 1188760, 1828860, 2093672, 2885342, 3379568, 4347890, 5089220, 7134860, 7835684, 10700654, 12422758, 14837078, 16812466, 20404320
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor m^2 is to ensure that a(n) is an integer.
A345423(n) = m*mu where mu is the mean of the values of u.
The population standard deviation sqrt(s) appears to grow linearly with n.

Crossrefs

Cf. A345423, A345687, A345688, A345689, A345690, A345691.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345692(n):
    zlist = [z for z in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if z[2] == 1]
    return pvariance(len(zlist)*u for u, v, w in zlist)

A345693 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = u*x+v*y with u,v minimal; a(n) = m^2*s, where s is the population variance of the values of v and m is the number of such values.

Original entry on oeis.org

0, 2, 26, 72, 374, 516, 2064, 3634, 7706, 10472, 25832, 34298, 70946, 90106, 128664, 177428, 317024, 376150, 623276, 757856, 987038, 1189074, 1829210, 2094022, 2885790, 3380040, 4348400, 5089782, 7135460, 7836276, 10701330, 12423438, 14837870, 16813314, 20405200
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor m^2 is to ensure that a(n) is an integer.
A345424(n) = m*mu where mu is the mean of the values of v.
The population standard deviation sqrt(s) appears to grow linearly with n.

Crossrefs

Cf. A345423, A345687, A345688, A345689, A345690, A345691, A345692.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345693(n):
    zlist = [z for z in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if z[2] == 1]
    return pvariance(len(zlist)*v for u, v, w in zlist)

A345694 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = u*x+v*y with u,v minimal; a(n) = m^2*s, where s is the population variance of the values of |u| and m is the number of such values.

Original entry on oeis.org

0, 2, 12, 28, 124, 168, 696, 1254, 2800, 3734, 9684, 13282, 27576, 34818, 51828, 71660, 129380, 153172, 254624, 312716, 413774, 496600, 767976, 879284, 1219286, 1422992, 1845842, 2173556, 3043292, 3345884, 4556174, 5288806, 6365966, 7188082, 8786288, 9615066
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor m^2 is to ensure that a(n) is an integer.
A345429(n) = m*mu where mu is the mean of the values of |u|.
The population standard deviation sqrt(s) appears to grow linearly with n.

Crossrefs

Cf. A345429, A345687, A345688, A345689, A345690, A345691, A345692, A345693.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345694(n):
    zlist = [z for z in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if z[2] == 1]
    return pvariance(len(zlist)*abs(u) for u, v, w in zlist)

A345695 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = u*x+v*y with u,v minimal; a(n) = m^2*s, where s is the population variance of the values of |v| and m is the number of such values.

Original entry on oeis.org

0, 2, 10, 24, 110, 152, 656, 1198, 2714, 3632, 9512, 13082, 27274, 34474, 51416, 71168, 128704, 152430, 253648, 311636, 412538, 495234, 766258, 877438, 1217102, 1420616, 1843136, 2170622, 3039784, 3342200, 4551830, 5284110, 6360830, 7182594, 8780236, 9608714
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor m^2 is to ensure that a(n) is an integer.
A345430(n) = m*mu where mu is the mean of the values of |v|.
The population standard deviation sqrt(s) appears to grow linearly with n.

Crossrefs

Cf. A345430, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345695(n):
    zlist = [z for z in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if z[2] == 1]
    return pvariance(len(zlist)*abs(v) for u, v, w in zlist)

A345696 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = u*x+v*y with u,v minimal; a(n) = m^2*s, where s is the population variance of the values of u^2+v^2 and m is the number of such values.

Original entry on oeis.org

0, 0, 10, 28, 846, 1080, 13524, 28336, 101274, 130086, 526116, 796704, 2121646, 2676676, 5103216, 7545320, 16863936, 20080798, 39983568, 51986376, 78689204, 96323998, 175534714, 207346098, 324942572, 386288432, 560665370, 693425934, 1087095852, 1220707044
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor m^2 is to ensure that a(n) is an integer.
A345431(n) = m*mu where mu is the mean of the values of u^2+v^2.
s^(1/4) appears to grow linearly with n.

Crossrefs

Cf. A345431, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694, A345695.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345696(n):
    zlist = [z for z in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if z[2] == 1]
    return pvariance(len(zlist)*(u**2+v**2) for u, v, w in zlist)

A345724 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = n^4*s, where s is the population variance of the values of u+v.

Original entry on oeis.org

0, 0, 14, 48, 250, 452, 1578, 2816, 6120, 9556, 20220, 28476, 54596, 75092, 111050, 155120, 253852, 323792, 497054, 624700, 828476, 1049584, 1510824, 1792476, 2397166, 2924432, 3736358, 4469884, 5919800, 6804500, 8811122, 10401536, 12541844, 14621072, 17574850
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor n^4 is to ensure that a(n) is an integer.
A345428(n) = n^2*mu where mu is the mean of the values of u+v.
The population standard deviation sqrt(s) appears to grow linearly with n.

Crossrefs

Cf. A345428, A345687, A345688, A345689, A345690, A345691.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345724(n): return pvariance(n**2*(u+v) for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)))

A345725 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = u*x+v*y with u,v minimal; a(n) = m^2*s, where s is the population variance of the values of u+v and m is the number of such values.

Original entry on oeis.org

0, 0, 10, 28, 166, 224, 964, 1624, 3626, 4934, 12308, 15928, 33670, 42828, 62656, 85016, 154016, 181254, 301688, 364896, 480428, 580134, 901698, 1021274, 1412852, 1655336, 2149650, 2503910, 3518644, 3847556, 5247764, 6093004, 7339188, 8291404, 10135408, 11018524
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

Comments

The factor m^2 is to ensure that a(n) is an integer.
A345425(n) = m*mu where mu is the mean of the values of u+v.
The population standard deviation sqrt(s) appears to grow linearly with n.

Crossrefs

Cf. A345425, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694, A345695.

Programs

  • Python
    from statistics import pvariance
from sympy.core.numbers import igcdex
def A345725(n):
    zlist = [z for z in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if z[2] == 1]
    return pvariance(len(zlist)*(u+v) for u, v, w in zlist)
Showing 1-8 of 8 results.

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

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