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.

A188226 Smallest number having exactly n divisors of the form 8*k + 7.

Original entry on oeis.org

1, 7, 63, 315, 945, 1575, 3465, 19845, 10395, 17325, 26775, 127575, 45045, 266805, 190575, 155925, 135135, 2480625, 225225, 130203045, 405405, 1289925, 2168775, 1715175, 675675, 3898125, 3468465, 1576575, 3239775, 67798585575, 2027025, 16769025, 2297295, 20539575, 42170625, 27286875, 3828825, 117661005
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 26 2011

Keywords

Comments

A188172(a(n)) = n and A188172(m) <> n for m < a(n).

Links

Crossrefs

Cf. A004771, A141164, A188172.
Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), A343105 (i=3), A343106 (i=5), this sequence (i=7).

Programs

  • Haskell
    import Data.List  (elemIndex)
import Data.Maybe (fromJust)
a188226 n = a188226_list !! n
a188226_list =
   map (succ . fromJust . (`elemIndex` (map a188172 [1..]))) [0..]

Extensions

a(19)-a(35) from Nathaniel Johnston, Apr 06 2011
More terms from Bert Dobbelaere, Apr 09 2021

A343106 Smallest number having exactly n divisors of the form 8*k + 5.

Original entry on oeis.org

1, 5, 45, 315, 585, 2205, 2925, 14175, 9945, 17325, 28665, 178605, 45045, 190575, 240975, 143325, 135135, 3189375, 225225, 93002175, 405405, 1403325, 1715175, 2401245, 675675, 3583125, 3239775, 1576575, 3468465, 94918019805, 2027025, 15436575, 2297295, 11609325, 16769025, 27286875, 3828825, 42879375, 117661005
Offset: 0

Views

Author

Jianing Song, Apr 05 2021

Keywords

Comments

Smallest index of n in A188171.
a(n) exists for all n, since 5*3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 5*3^0, 5*3^2, ..., 5*3^(2n-2). This actually gives an upper bound (which is too far from reality when n is large) for a(n).

Examples

			a(4) = 585 since it is the smallest number with exactly 4 divisors congruent to 5 modulo 8, namely 5, 13, 45 and 585.

Links

Crossrefs

Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), A343105 (i=3), this sequence (i=5), A188226 (i=7).
Cf. A188171.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
a(n) = for(k=1, oo, if(res(k,8,5)==n, return(k)))

Formula

a(2n-1) <= 3^(2n-2) * 35, since 3^(2n-2) * 35 has exactly 2n-1 divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^(2n-2) * 5, 3^1 * 7, 3^3 * 7, ..., 3^(2n-3) * 7.
a(2n) <= 3^(n-1) * 455, since 3^(n-1) * 455 has exactly 2n divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^b * 5, 3^1 * 7, 3^3 * 7, ..., 3^a * 7, 3^0 * 13, 3^2 * 13, ..., 3^b * 13, 3^1 * 455, 3^3 * 455, ... 3^a * 455, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.

Extensions

More terms from Bert Dobbelaere, Apr 09 2021

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

A343104 Smallest number having exactly n divisors of the form 8*k + 1.

Original entry on oeis.org

1, 9, 81, 153, 891, 1377, 8019, 3825, 11025, 15147, 88209, 31977, 354375, 99225, 121275, 95931, 7144929, 187425, 893025, 287793, 1403325, 1499553, 1715175, 675675, 1091475, 6024375, 1576575, 1686825, 72335025, 2027025, 2264802453041139, 2297295, 11609325, 121463793, 9823275
Offset: 1

Views

Author

Jianing Song, Apr 05 2021

Keywords

Comments

Smallest index of n in A188169.
a(n) exists for all n, since 3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 3^0, 3^2, ..., 3^(2n-2). This actually gives an upper bound for a(n).
From David A. Corneth, Apr 05 2021: (Start)
All terms are odd since if a term is even then the odd part has the same number of such divisors.
No a(2*k + 1) is divisible by a prime congruent to 1 (mod 8).
If for some k, A188169(k) > m then A188169(k*t) > m for all t > 0. This can be used to trim searches when looking for some a(m).
If gcd(k, m) = 1 then A188169(k) * A188169(m) <= A188169(k*m) (End)

Examples

			a(4) = 153 since it is the smallest number with exactly 4 divisors congruent to 1 modulo 8, namely 1, 9, 17 and 153.

Crossrefs

Smallest number having exactly n divisors of the form 8*k + i: this sequence (i=1), A343105 (i=3), A343106 (i=5), A188226 (i=7).
Cf. A188169.
Cf. A007519, A007520, A007521, A007522.

Programs

  • PARI
    res(n,a,b) = sumdiv(n, d, (d%a) == b)
a(n) = if(n>0, for(k=1, oo, if(res(k,8,1)==n, return(k))))

Formula

a(2n-1) <= 3^(2n-2) * 11, since 3^(2n-2) * 11 has exactly 2n-1 divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^(2n-2), 3^1 * 11, 3^3 * 11, ..., 3^(2n-3) * 11.
a(2n) <= 3^(n-1) * 187, since 3^(n-1) * 187 has exactly 2n divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^b, 3^0 * 17, 3^2 * 17, ..., 3^b * 17, 3^1 * 11, 3^3 * 11, ..., 3^a * 11, 3^1 * 187, 3^3 * 187, ... 3^a * 187, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.

Extensions

More terms from David A. Corneth, Apr 06 2021

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

A343135 Positions of records in A188170.

Original entry on oeis.org

1, 3, 27, 99, 297, 891, 1683, 5049, 15147, 31977, 95931, 225225, 287793, 675675, 1091475, 1576575, 2027025, 2297295, 3828825, 6185025, 11486475, 18555075, 26801775, 34459425, 43648605, 72747675, 117515475, 218243025, 352546425, 509233725, 654729075, 1003917915, 1527701175, 3011753745
Offset: 1

Views

Author

David A. Corneth, Apr 06 2021

Keywords

Examples

			297 is in the sequence as A188170(297) = 4 via the divisors 3, 11, 27 and 99 and no positive integer < 297 has at least four such (3 (mod 8)) divisors.

Crossrefs

Cf. A188170, A343105.
Showing 1-8 of 8 results.

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