A003136 -id:A003136 - cp's OEIS frontend

A260682 Löschian numbers (A003136) of the form 6*k+1.

1, 7, 13, 19, 25, 31, 37, 43, 49, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 169, 175, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 289, 301, 307, 313, 325, 331, 337, 343, 349, 361, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 475, 481, 487, 499

Offset: 1

Joerg Arndt, Nov 15 2015

nonn

Odd terms of A202822, which lists Löschian numbers of the form 3*k+1. - Altug Alkan, Nov 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.

Cf. A003136, A016921, A202822.

a260682 n = a260682_list !! (n-1)
a260682_list = filter ((== 1) . flip mod 6) a003136_list
-- Reinhard Zumkeller, Nov 16 2015

nn = 25; Select[Union[Flatten[Table[x^2 + x*y + y^2, {x, 0, nn}, {y, 0, x}]]], Mod[#, 6] == 1 && # <= nn^2&] (* Jean-François Alcover, Jul 21 2018, after T. D. Noe *)

is(n)=(n%6==1)&&#bnfisintnorm(bnfinit(z^2+z+1), n);
select(n->is(n), vector(500,j,j))

x='x+O('x^500); p=eta(x)^3/eta(x^3); for(n=0, 499, if(polcoeff(p, n) != 0 && n%6==1, print1(n, ", "))) \\ Altug Alkan, Nov 15 2015

isok(n) = if( n<1 || (n%3 == 0), 0, 0 != sumdiv( n, d, kronecker( -3, d))) && n%2==1;
for(n=0, 500, if(isok(n), print1(n", "))) \\ Altug Alkan, Nov 15 2015

list(lim)=my(v=List(), y, t); for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, if(t%6==1, listput(v, t)); y++)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017

A364443 a(n) is the number of integers k of the form x^2 + x*y + y^2 (A003136) with n^2 < k < (n+1)^2.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 5, 4, 6, 5, 6, 8, 7, 8, 7, 9, 9, 11, 10, 10, 11, 10, 13, 12, 13, 13, 13, 14, 13, 16, 16, 16, 14, 16, 17, 16, 18, 20, 19, 19, 19, 19, 21, 20, 22, 21, 21, 22, 22, 24, 25, 21, 24, 25, 24, 27, 27, 25, 29, 26, 28, 26, 27, 29, 29, 30, 28, 29, 32, 31

Offset: 0

Hugo Pfoertner, Aug 05 2023

nonn

a(n) is the number of circles centered at (0,0) that pass through grid points of the hexagonal lattice that intersect the interior of an interval n < x < n+1 on the x-axis.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
IBM Research, Circles on a triangular grid, Ponder This Challenge December 2023.
Hugo Pfoertner, Table of n, a(n) for n = 0..500000

Cf. A002324, A004016, A077773, A357112, A364729, A364730, A364731, A364732.

is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3);
for (k=0, 75, my (k1=k^2+1, k2=k^2+2*k, m=0); for (j=k1, k2, m+=is_a003136(j)); print1(m,", "))

from sympy import factorint
def A364443(n): return sum(1 for k in range(n**2+1,(n+1)**2) if not any(e&1 for p, e in factorint(k).items() if p % 3 == 2)) # Chai Wah Wu, Aug 07 2023

A024610 Position of n^2 in A003136.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 21, 26, 32, 37, 44, 50, 57, 66, 74, 83, 91, 101, 111, 123, 134, 145, 157, 168, 182, 195, 209, 223, 237, 252, 266, 283, 300, 317, 332, 349, 367, 384, 403, 424, 444, 464, 484, 504, 526, 547, 570, 592, 614, 637, 660, 685, 711, 733, 758, 784

Offset: 0

Clark Kimberling

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A000290 (n^2), A003136.

More terms from Naohiro Nomoto, Nov 30 2001
Term 0 added by Ray Chandler, Jan 28 2009
Offset corrected by Sean A. Irvine, Jul 17 2019

A198726 Number of partitions of n into positive Loeschian numbers (cf. A003136).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 6, 7, 9, 11, 13, 17, 21, 24, 29, 37, 42, 49, 60, 70, 82, 96, 111, 129, 152, 173, 199, 234, 266, 302, 349, 399, 451, 515, 585, 661, 752, 847, 954, 1081, 1215, 1359, 1531, 1719, 1917, 2147, 2400, 2675, 2985, 3322, 3690, 4110, 4563, 5048, 5603

Offset: 0

Reinhard Zumkeller, Oct 30 2011

nonn

			a(10) = #{9+1, 7+3, 7+1+1+1, 4+4+1+1, 4+3+3, 4+3+1+1+1, 4+6x1, 3+3+3+1, 3+3+1+1+1+1, 3+7x1, 10x1} = 11;
a(11) = #{9+1+1, 7+4, 7+3+1, 7+1+1+1+1, 4+4+3, 4+4+1+1+1, 4+3+3+1, 4+3+4x1, 4+7x1, 3+3+3+1+1, 3+3+5x1, 3+8x1, 11x1} = 13;
a(12) = #{12, 9+3, 9+1+1+1, 7+4+1, 7+3+1+1, 7+5x1, 4+4+4, 4+4+3+1, 4+4+4x1, 4+3+3+1+1, 4+3+5x1, 4+8x1, 3+3+3+3, 3+3+3+1+1+1, 3+3+6x1, 3+9x1, 12x1} = 17.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A003136, A198727.

import Data.MemoCombinators (memo2, list, integral)
a198726 n = a198726_list !! n
a198726_list = f 0 [] $ tail a003136_list where
   f u vs ws'@(w:ws) | u < w = (p' vs u) : f (u + 1) vs ws'
                     | otherwise = f u (vs ++ [w]) ws
   p' = memo2 (list integral) integral p
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m
-- Reinhard Zumkeller, Nov 16 2015, Oct 30 2011

A198727 Number of partitions of n into distinct positive Loeschian numbers (cf. A003136).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 0, 2, 2, 1, 2, 2, 3, 4, 3, 2, 6, 6, 1, 5, 9, 6, 5, 9, 9, 9, 11, 8, 13, 17, 11, 12, 22, 19, 13, 23, 25, 22, 26, 28, 30, 37, 34, 31, 47, 45, 35, 50, 61, 52, 56, 69, 68, 76, 74, 72, 95, 100, 82, 100, 130, 112, 113, 139, 144, 149, 154, 156, 183

Offset: 0

Reinhard Zumkeller, Oct 30 2011

nonn

			a(20) = #{19+1, 16+4, 16+3+1, 13+7, 13+4+3, 12+7+1, 12+4+3+1, 9+7+4, 9+7+3+1} = 9;
a(21) = #{21, 16+4+1, 13+7+1, 13+4+3+1, 12+9, 9+7+4+1} = 6;
a(22) = #{21+1, 19+3, 13+9, 12+9+1, 12+7+3} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A003136, A198726.

import Data.MemoCombinators (memo2, list, integral)
a198727 n = a198727_list !! n
a198727_list = f 0 [] $ tail a003136_list where
   f u vs ws'@(w:ws) | u < w = (p' vs u) : f (u + 1) vs ws'
                     | otherwise = f u (vs ++ [w]) ws
   p' = memo2 (list integral) integral p
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m = if m < k then 0 else p' ks (m - k) + p' ks m
-- Reinhard Zumkeller, Nov 16 2015, Oct 30 2011

A230656 Position of the circles with center at a lattice point that contain record numbers of lattice points, in the list of Loeschian numbers A003136.

Original entry on oeis.org

1, 2, 5, 21, 34, 186, 454, 1927, 2746, 11128, 30906, 70762, 349068, 830170, 2278644, 13219495, 27373350, 87615127

Offset: 0

Hugo Pfoertner, Oct 27 2013

			a(3) = 21 because A230655(3) = A003136(21) = 49.

Cf. A003136 (Loeschian numbers), A230655, A075880 (similar sequence for square lattice)

a(n) = k for which A230655(n) = A003136(k).

A264732 Löschian numbers (A003136) which are the sum of 2 nonzero squares.

Original entry on oeis.org

13, 25, 37, 52, 61, 73, 97, 100, 109, 117, 148, 157, 169, 181, 193, 208, 225, 229, 241, 244, 277, 289, 292, 313, 325, 333, 337, 349, 373, 388, 397, 400, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 625, 628, 637, 657, 661, 673, 676, 709, 724, 733

Offset: 1

Altug Alkan, Nov 22 2015

nonn

n is in the sequence iff 4*n is.
If a(n) is a prime number, a(n) mod 12 = 1.
Prime terms of sequence are listed in A068228 that lists generalized cuban primes (A007645) which are the sum of 2 nonzero squares.
Also positive numbers of the form x^2 - 3*y^2 (A084916) that are the sum of 2 nonzero squares. - Frank M Jackson, Oct 13 2019

			a(1) = 13 because 13 = 3^2 + 3*1 + 1^2 = 3^2 + 2^2.
a(2) = 25 because 25 = 5^2 + 5*0 + 0^2 = 4^2 + 3^2.
a(3) = 37 because 37 = 4^2 + 4*3 + 3^2 = 6^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000404, A003136, A068228, A084916.

Select[Range@750, Length[PowersRepresentations[#, 2, 2] /. {0, }->Nothing]>0 && Reduce[#==x^2+x*y+y^2, {x, y}, Integers]=!=False &] (* _Frank M Jackson, Oct 13 2019 *)

isok(n) = { for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=1, 1e3, if( is(n) && isok(n), print1(n, ", ")))

A328793 Least k such that there exists an equilateral triangle of side length sqrt(A003136(n)) with vertices in an equilateral triangular array of points with k rows.

Original entry on oeis.org

2, 4, 3, 5, 4, 7, 6, 5, 8, 7, 6, 10, 9, 8, 7, 11, 10, 9, 13, 8, 11, 10, 14, 13, 9, 12, 11, 16, 15, 14, 10, 13, 12, 16, 15, 11, 14, 19, 18, 13, 17, 16, 12, 15, 20, 19, 14, 17, 13, 16, 21, 20, 19, 15, 18, 14, 22, 17, 21, 20, 16, 19, 25, 24, 15, 18, 22, 21, 17

Offset: 1

Peter Kagey, Oct 27 2019

a(A024610(n) - 1) = n + 1 for all n > 0, and

a(A024610(n) + k) > n + 1 for all n > 0, k >= 0.

			For n = 2, there exists an equilateral triangle with side length sqrt(A003136(2)) = sqrt(3) and vertices on equilateral triangular array with a(2) = 4 rows:
     o
    * o
   o o *
  o * o o.
However there is no equilateral triangle of side length sqrt(3) with vertices on a smaller triangular array with three or fewer rows.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A003136, A024610, A328787.

A155563 Intersection of A001481 and A003136: N = a^2 + b^2 = c^2 + 3d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 9, 13, 16, 25, 36, 37, 49, 52, 61, 64, 73, 81, 97, 100, 109, 117, 121, 144, 148, 157, 169, 181, 193, 196, 208, 225, 229, 241, 244, 256, 277, 289, 292, 313, 324, 325, 333, 337, 349, 361, 373, 388, 397, 400, 409, 421, 433, 436, 441, 457, 468, 481, 484

Offset: 1

M. F. Hasler, Jan 24 2009

Contains A155561 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Ron Lifshitz, Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Mod. Phys. 69, 1181 (1997). This sequence coincides with the row N = 12 of Table VII.

Cf. A000290, A001481, A003136, A155561.

isA155563(n,/* use optional 2nd arg to get other analogous sequences */c=[3,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,500, isA155563(n) & print1(n","))

is(n)=(n==0) || (#bnfisintnorm(bnfinit(z^2+z+1), n) && #bnfisintnorm(bnfinit(z^2+1), n));
select(n->is(n), vector(1500,j,j-1)) \\ Joerg Arndt, Jan 11 2015

from itertools import count, islice
from sympy import factorint
def A155563_gen(): # generator of terms
    return filter(lambda n: all(e & 1 == 0 or (p & 3 != 3 and p % 3 < 2) for p, e in factorint(n).items()),count(0))
A155563_list = list(islice(A155563_gen(),30)) # Chai Wah Wu, Jun 27 2022

A155564 Intersection of A002479 and A003136: N = a^2 + 2b^2 = c^2 + 3d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 3, 4, 9, 12, 16, 19, 25, 27, 36, 43, 48, 49, 57, 64, 67, 73, 75, 76, 81, 97, 100, 108, 121, 129, 139, 144, 147, 163, 169, 171, 172, 192, 193, 196, 201, 211, 219, 225, 228, 241, 243, 256, 268, 283, 289, 291, 292, 300, 304, 307, 313, 324, 331, 337, 361, 363

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155574 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155564(n,/* use optional 2nd arg to get other analogous sequences */c=[3,2]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,500, isA155564(n) & print1(n","))

cp's OEIS Frontend

A260682 Löschian numbers (A003136) of the form 6*k+1.

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A364443 a(n) is the number of integers k of the form x^2 + x*y + y^2 (A003136) with n^2 < k < (n+1)^2.

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A024610 Position of n^2 in A003136.

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A198726 Number of partitions of n into positive Loeschian numbers (cf. A003136).

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A198727 Number of partitions of n into distinct positive Loeschian numbers (cf. A003136).

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A230656 Position of the circles with center at a lattice point that contain record numbers of lattice points, in the list of Loeschian numbers A003136.

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A264732 Löschian numbers (A003136) which are the sum of 2 nonzero squares.

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A328793 Least k such that there exists an equilateral triangle of side length sqrt(A003136(n)) with vertices in an equilateral triangular array of points with k rows.

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A155563 Intersection of A001481 and A003136: N = a^2 + b^2 = c^2 + 3d^2 for some integers a,b,c,d.

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