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 1-7 of 7 results.

A374158 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 3*y^2 = k.

Original entry on oeis.org

0, 4, 91, 28, 196, 31213, 364, 9604, 53599, 2548, 470596
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2024

Keywords

Comments

a(n) is the smallest nonnegative k such that A092573(k) = n.
a(11) <= 3672178237.
a(12) = 6916.
a(13) = 33124.
a(14) = 29059303.
a(15) = 124852.
a(16) = 1983163.
a(18) = 48412.
a(20) = 18384457.
a(21) = 6117748.
a(22) = 1623076.
a(24) = 214396.
a(27) = 629356.
a(28) = 900838393.
a(31) = 79530724.
a(32) = 85276009.
a(37) = 274299844.
a(42) = 116237212.
a(60) = 73537828.
a(67) = 585930436.
From Chai Wah Wu, Jun 29-30 2024: (Start)
a(30) = 2372188.
a(36) = 1500772.
a(40) = 11957764.
a(45) = 30838444.
a(48) = 7932652.
a(54) = 19510036.
a(72) = 55528564.
(End)

Examples

			   n | a(n)
-----+---------------------------
   1 |      4 = 2^2.
   2 |     91 = 7 * 13.
   3 |     28 = 2^2 * 7.
   4 |    196 = 2^2 * 7^2.
   5 |  31213 = 7^4 * 13.
   6 |    364 = 2^2 * 7 * 13.
   7 |   9604 = 2^2 * 7^4.
   8 |  53599 = 7 * 13 * 19 * 31.
   9 |   2548 = 2^2 * 7^2 * 13.
  10 | 470596 = 2^2 * 7^6.

Crossrefs

Cf. A328151, A343105, A374159, A374160, A374161.
Cf. A002476, A092573.

Programs

  • Python
    from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374158(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+3*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 29 2024

A374160 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 11*y^2 = k.

Original entry on oeis.org

0, 12, 60, 180, 540, 1620, 2700, 8100, 12420, 20700, 37260, 1180980, 62100, 476100, 335340, 186300, 310500, 1822500, 558900, 53144100, 931500, 1676700, 4284900, 324860625, 1925100, 4657500, 244462860, 12854700, 8383500
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2024

Keywords

Comments

a(n) is the smallest nonnegative k such that A374017(k) = n.
a(30) = 5775300.
a(31) = 38564100.
a(32) = 9625500.
a(33) = 135812700.
a(35) = 41917500.
a(36) = 17325900.
a(37) = 107122500.
a(40) = 28876500.

Crossrefs

Cf. A328151, A343105, A374158, A374159, A374161.
Cf. A374017.

Programs

  • Python
    from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374160(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+11*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 30 2024

A374159 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 7*y^2 = k.

Original entry on oeis.org

0, 8, 32, 128, 352, 704, 1408, 2816, 5632, 11264, 16192, 45056, 32384, 123904, 64768, 178112, 129536, 2883584, 259072, 1982464, 469568, 712448, 1036288, 184549376, 939136, 21551552, 4145152, 2849792, 1878272
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2024

Keywords

Comments

a(n) is the smallest nonnegative k such that A216511(k) = n.
Conjecture: All terms are multiple of a(1) = 8.
a(30) = 5165248.
a(31) = 16386304.
a(32) = 3756544.
a(33) = 11399168.
a(34) = 66322432.
a(35) = 86206208.
a(36) = 7513088.

Crossrefs

Cf. A328151, A343105, A374158, A374160, A374161.
Cf. A216511.

Programs

  • Python
    from itertools import count
from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374159(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+7*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 30 2024

A374161 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 19*y^2 = k.

Original entry on oeis.org

0, 20, 140, 700, 1540, 17500, 7700, 122500, 26180, 53900, 192500, 7035875, 130900, 592900, 4812500, 1347500, 602140, 150062500, 916300
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2024

Keywords

Crossrefs

Cf. A328151, A343105, A374158, A374159, A374160.

Programs

  • Python
    from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374161(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+19*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 30 2024

A374090 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + x*y + y^2 = k.

Original entry on oeis.org

0, 3, 7, 147, 91, 7203, 637, 352947, 1729, 24843, 31213, 847425747, 12103, 41523861603, 405769, 1217307, 53599, 99698791708803, 157339, 4885240793731347, 593047, 59648043
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2024

Keywords

Comments

a(n) is the smallest nonnegative k such that A374088(k) = n.
From Chai Wah Wu, Jun 28 2024: (Start)
If x <> y and x^2 + x*y + y^2 = a(n), then (x, y) and (y, x) both count as solutions. Therefore if a(n) exists, then a(n) is of the form 3*m^2 if and only if n is odd. This also implies that a(2*n) = A374094(n).
a(25) = 205724883.
a(27) = 8968323.
a(33) = 143214951243.
a(35) = 10080519267.
a(45) = 439447827.
a(49) = 1703607756123.
a(63) = 21532943523.
a(75) = 74266682763.
a(81) = 8618558403.
a(135) = 422309361747.
(End)
From David A. Corneth, Jun 29 2024: (Start)
a(19) <= 3*7^18.
a(22) <= 3672178237.
a(24) = 375193.
a(26) = 2989441 <= 179936733613.
a(28) = 29059303.
a(30) = 7709611.
a(32) = 1983163.
a(34) <= 432028097404813.
a(36) = 4877509.
Conjecture: Let q_i be the i-th prime of the form 3*k + 1 and let m = Prod_{j=1, t} b_j, a factorization of m into factors > 1.
Let f(m) = Prod_{j = 1..t} q_i^(b_(t+1-j)-1).
Then for even n we have a(n) = min(f(n), f(n+1))
and for odd n we have a(n) = 3*f(n).
Example for n = 22 we might factor 22 = 11*2. The first two primes of the form 3*k + 1 are 7 and 13. So we would have a(22) = min(7^10*13, 7^22).
a(14) = min(f(14), f(15)) = min(7^6 * 13, 7^4 * 13^2) = 405769. (End)

Crossrefs

Cf. A002476, A328151, A374091.
Cf. A003136, A374088, A374094.

Programs

  • Python
    from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374090(n): return next(m for m in (3*k**2 if n&1 else k for k in count(0)) if sum(1 for d in diop_quadratic(x*(x+y)+y**2-m) if d[0]>0 and d[1]>0) == n) # Chai Wah Wu, Jun 28 2024

Formula

a(2*n) = A374094(n).

Extensions

a(11), a(13) from Chai Wah Wu, Jun 28 2024
a(17) from Bert Dobbelaere, Jun 28 2024
a(19) from Bert Dobbelaere, Jun 30 2024

A374091 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + 3*x*y + y^2 = k.

Original entry on oeis.org

0, 5, 11, 605, 209, 73205, 2299, 8857805, 6061, 218405, 278179
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2024

Keywords

Comments

a(n) is the smallest nonnegative k such that A374089(k) = n.
a(n) is of the form 5*k^2 if and only if n is odd.
a(12) = 66671.
a(14) = 5285401.
a(15) = 26427005.
a(16) = 187891.
a(18) = 1266749.
a(20) = 8067191.
a(21) = 3197667605. - Chai Wah Wu, Jun 29 2024
a(24) = 2066801.
a(26) = 36735721.
a(27) = 183678605. - Chai Wah Wu, Jun 28 2024

Crossrefs

Cf. A328151, A374090.
Cf. A374089, A374095.

Formula

a(2*n) = A374095(n).
a(n) <= 5*11^(n-1) for all n >= 1. - Jason Yuen, Jun 29 2024

A355814 Smallest value t such that 1/s^2 + 1/t^2 = 1/p^2 + 1/q^2 has exactly n solutions (p,q) where p,q < t; or -1 if no such t exists.

Original entry on oeis.org

35, 55, 210, 240, 595, 360, 560, 504, 630, 720, 1295, 1848, 1890, 1386, 1680, 2640, 2520, 3024, 5600, 3960, 2730, 4680, 6160, 8775, 9450, 5850, 5460, 5544, 9520, 15470, 5040, 7920, 9240, 25740, 10710, 9360, 13860, 13104, 8190, 17550, 10920, 18720, 15120, 22176
Offset: 1

Views

Author

Jianing Song, Jul 18 2022

Keywords

Comments

Terms beyond a(11) = 1295 other than a(14) = 1386, if not equal to -1, are greater than 1500.
Conjecture: a(n) is divisible by 35 for odd n.

Examples

			t = 35: (s,p,q) = (5,7,7);
t = 55: (s,p,q) = (10,11,22),(10,22,11);
t = 210: (s,p,q) = (30,42,42),(95,114,133),(95,133,114);
t = 240: (s,p,q) = (70,84,112),(70,112,84),(108,135,144),(108,144,135);
t = 595: (s,p,q) = (85,91,221),(85,119,119),(85,221,91),(210,238,357),(210,357,238);
t = 360: (s,p,q) = (20,24,36),(20,36,24),(30,40,45),(30,45,40),(105,126,168),(105,168,126);
t = 560: (s,p,q) = (45,48,126),(70,80,140),(80,112,112),(45,126,48),(70,140,80),(252,315,336),(252,336,315);
t = 504: (s,p,q) = (42,56,63),(54,56,189),(42,63,56),(63,72,126),(63,126,72),(112,144,168),(112,168,144),(54,189,56);
t = 630: (s,p,q) = (35,42,63),(35,63,42),(56,63,120),(56,120,63),(90,126,126),(140,180,210),(140,210,180),(285,342,399),(285,399,342);
t = 720: (s,p,q) = (40,48,72),(40,72,48),(60,80,90),(60,90,80),(165,176,396),(210,252,336),(210,336,252),(165,396,176),(324,405,432),(324,432,405).

Crossrefs

Cf. A328151, A355812, A355813.

Programs

  • PARI
    b(n) = my(v=[;],r); for(p=1, n-1, for(q=1, n-1, r=1/(1/p^2+1/q^2-1/n^2); if(r==r\1 && issquare(r), v=concat(v,[p;q])))); v
search_up_to(Max,lim) = my(v=vector(Max,i,-1),num); for(n=1, lim, if((num=#b(n))>0 && num<=Max && v[num]==-1, v[num]=n)); v

Extensions

a(12)-a(29) from Bert Dobbelaere, Jul 19 2022
More terms from Jinyuan Wang, Jan 25 2025
Showing 1-7 of 7 results.

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