A024614 -id:A024614 - cp's OEIS frontend

A024615 Positions of even numbers in A024614.

3, 8, 13, 15, 22, 24, 29, 32, 34, 40, 42, 47, 52, 54, 57, 62, 67, 69, 71, 78, 79, 81, 85, 90, 97, 99, 104, 109, 113, 115, 118, 119, 123, 130, 133, 136, 139, 145, 151, 153, 157, 161, 162, 168

Offset: 1

Clark Kimberling

nonn

A024616 Positions of odd numbers in A024614.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 48, 49, 50, 51, 53, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 70, 72, 73, 74, 75, 76, 77, 80, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 98, 100, 101

Offset: 1

Clark Kimberling

nonn

A024617 Position of n^2 + n + 1 in A024614.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 31, 37, 43, 50, 58, 65, 73, 82, 92, 102, 111, 122, 132, 144, 155, 167, 181, 194, 207, 222, 237, 252, 267, 283, 297, 314, 333, 349, 367, 385, 404, 422, 444, 466, 483, 505, 524, 546, 568, 591, 615, 637, 662, 684, 706, 733, 759, 784

Offset: 1

Clark Kimberling

nonn

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 19 2008

A024618 Positions of primes in A024614.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 12, 17, 19, 20, 23, 27, 28, 30, 35, 38, 41, 43, 44, 49, 53, 55, 58, 61, 63, 65, 72, 74, 76, 82, 84, 88, 91, 93, 98, 100, 101, 105, 108, 111, 114, 116, 121, 122, 128, 131, 137

Offset: 1

Clark Kimberling

nonn

A353198 Intersection of A000404 and A024614.

Original entry on oeis.org

13, 37, 52, 61, 73, 97, 109, 117, 148, 157, 169, 181, 193, 208, 229, 241, 244, 277, 292, 313, 325, 333, 337, 349, 373, 388, 397, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 628, 637, 657, 661, 673, 676, 709, 724, 733, 757, 769, 772, 793, 829, 832, 853, 873, 877

Offset: 1

Steven Lu, May 16 2022

nonn

A000404 lists the numbers that are the squares of Euclidean distances between two lattice points on a square grid, the segment between which is not parallel to either axis, and thus the square grid could be divided into such number of parts symmetrically, periodically and nontrivially. Similarly, A024614 lists the numbers that are the squares of Euclidean distances between two lattice points on a hexagonal grid, the segment between which is not parallel to any of the axes.
Thus this sequence lists the numbers into which both the square grid and hexagonal one could be divided.
This sequence contains all the prime numbers of the form p = 12*k + 1 and, for each such prime, all composites of the form c = j^2*p.

			The square grid (or the Gaussian integers) can be divided into 13 parts, where the k-th part consists of grid points of the form (k + 3*n + 2*m, 2*n - 3*m) where n, m are integers. Similarly the hexagonal grid can be also divided into 13 parts, where the k-th part consists of points of the form (k + 7/2*n + m, sqrt(3)/2*n + 2*sqrt(3)*m).

Cf. A000404, A024614.

Select[Intersection[
  Sort[DeleteDuplicates[
    Flatten[Table[i^2 + j^2, {i, 1000}, {j, 1000}]]]],
  Sort[DeleteDuplicates[
    Flatten[Table[i^2 + i j + j^2, {i, 1000}, {j, 1000}]]]]], # <=
   10000 &]

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613

Offset: 1

N. J. A. Sloane, Mira Bernstein and Robert G. Wilson v

Also, odd primes p such that -3 is a square mod p. - N. J. A. Sloane, Dec 25 2017
Equivalently, primes of the form p = (x^3 - y^3)/(x - y). If x=y+1 we get the cuban primes A002407, which is therefore a subsequence.
These are not to be confused with the Eisenstein primes, which are the primes in the ring of integers Z[w], where w = (-1+sqrt(-3))/2. The present sequence gives the rational primes which are also Eisenstein primes. - N. J. A. Sloane, Feb 06 2008
Also primes of the form x^2+3y^2 and, except for 3, x^2+xy+7y^2. See A140633. - T. D. Noe, May 19 2008
Conjecture: this sequence is Union(A002383,A162471). - Daniel Tisdale, Jul 04 2009
Primes p such that antiharmonic mean B(p) of the numbers k < p such that gcd(k, p) = 1 is not integer, where B(p) = A053818(p) / A023896(p) = A175505(p) / A175506(p) = (2p - 1) / 3. Primes p such that A175506(p) > 1. Subsequence of A179872. Union a(n) + A179891 = A179872. Example: a(6) = 37 because B(37) = A053818(37) / A023896(37) = A175505(37) / A175506(37) = 16206 / 666 = 73 / 3 (not integer). Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Subsequence of Loeschian numbers, cf. A003136 and A024614; A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011
Primes such that there exist a unique x, y, with 1 < x <= y < p, x + y == 1 (mod p) and x * y == 1 (mod p). - Jon Perry, Feb 02 2014
The prime factors of A002061. - Richard R. Forberg, Dec 10 2014
This sequence gives the primes p which solve s^2 == -3 (mod 4*p) (see Buell, Proposition 4.1., p. 50, for Delta = -3). p = 2 is not a solution. x^2 == -3 (mod 4) has solutions for all odd x. x^2 == -3 (mod  p) has for odd primes p, not 3, the solutions of Legendre(-3|p) = +1 which are p == {1, 7} (mod 12). For p = 3 the representative solution is x = 0. Hence the solution of s^2 == -3 (mod 4*p) are the odd primes p = 3 and p == {1, 7} (mod 12) (or the primes p = 0, 1 (mod 3)). - Wolfdieter Lang, May 22 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 50.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

T. D. Noe, Table of n, a(n) for n = 1..1000
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Eisenstein Integer.

Subsequence of A003136.
Subsequences include A002407, A002648, and A201477.
Apart from initial term, same as A045331.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2).
Cf. A000040, A003627.
Primes in A003136 and A034017.

a007645 n = a007645_list !! (n-1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
-- Reinhard Zumkeller, Jul 11 2013, Oct 30 2011

select(isprime,[3, seq(6*k+1, k=1..1000)]); # Robert Israel, Dec 12 2014

Join[{3},Select[Prime[Range[150]],Mod[#,3]==1&]] (* Harvey P. Dale, Aug 21 2021 *)

forprime(p=2,1e3,if(p%3<2,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

p == 0 or 1 (mod 3).
{3} UNION A002476. - R. J. Mathar, Oct 28 2008
A007645 UNION A003627 = A000040. - Juri-Stepan Gerasimov, Jan 28 2010

Entry revised by N. J. A. Sloane, Jan 29 2013

A024606 Numbers of form x^2 + xy + y^2 with distinct x and y > 0.

Original entry on oeis.org

7, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 63, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 193, 196, 199, 201, 208, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247

Offset: 1

Clark Kimberling

Alternatively, numbers expressible in more than one way as i^2 - ij + j^2, where 1 <= i <= j or 1 <= i < j. The following argument shows that the conditions i <= j or i < j are here equivalent. Note first that i^2 - ij + j^2 = (j - i)^2 - (j - i)*j + j^2, so the only non-duplicated values i^2 - ij + j^2 with 1 <= i < j are when j = 2i, whence i^2 - ij + j^2 = 3i^2. On the other hand, the values with i = j are j^2. There are no integer solutions to 3i^2 = j^2 with i >= 1. - Franklin T. Adams-Watters, May 03 2006
Numbers whose prime factorization contains at least one prime congruent to 1 mod 6 and any prime factor congruent to 2 mod 3 has even multiplicity. - Franklin T. Adams-Watters, May 03 2006
This is a subsequence of Loeschian numbers A003136, closed under multiplication. Its primitive elements are those with exactly one prime factor of form 6k + 1 with multiplicity one (A232436). - Jean-Christophe Hervé, Nov 22 2013
a(1)*a(2)*a(3) = 1729, the Hardy-Ramanujan taxicab number. 1729 is then in the sequence, by the argument of the preceding comment. - Jean-Christophe Hervé, Nov 24 2013
1729 is also the least term that can be written in 4 distinct ways in the given form. Sequence A024614 does not include the restriction x != y, it is the disjoint union of this sequence and A033428 (i.e., 3*x^2) (without 0). - M. F. Hasler, Mar 05 2018

			a(1) = 7 = 1^2 + 2 + 2^2.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, A Multisection of q-Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003136, A004016, A024614, A074628, A088534, A118886, A232436.

Take[Union[Flatten[Table[x^2 + x*y + y^2, {x, 15}, {y, x - 1}]]], 60] (* Robert G. Wilson v, Nov 24 2013 *)

for(k=1,247,my(a088534=sum(x=0,sqrt(k\3),sum(y=max(x,sqrtint(k-x^2)\2),sqrtint(k-2*x^2),x^2+x*y+y^2==k)),a004016d6=sumdiv(k,d,(d%3==1)-(d%3==2)));if(a088534!=a004016d6,print1(k,", "))) \\ Hugo Pfoertner, Sep 22 2019

A004016(a(n)) >= 12. - Jean-Christophe Hervé, Nov 24 2013

Definition modified by Alonso del Arte and Jean-Christophe Hervé, Nov 25 2013

A300419 Smallest nonnegative number k such that k can be written in exactly n ways as x^2 + xy + y^2 where x and y are positive integers, with x >= y.

Original entry on oeis.org

0, 3, 91, 637, 1729, 24843, 12103, 405769, 53599, 157339, 593047, 59648043, 375193, 2989441, 8968323, 7709611, 1983163, 3360173089, 4877509, 2339177536969, 18384457, 377770939, 146482609, 439447827, 13882141, 1302924259

Offset: 0

Altug Alkan, M. F. Hasler and Robert G. Wilson, Mar 05 2018

Except a(0) and a(1), all terms are in A118886.
First positive square term of this sequence is a(7) = 405769 = a(3)^2.
a(3), a(7) = a(3)^2 and a(13) = a(4)^2 are also the sum of two nonzero squares in exactly one way.
a(18) = 4877509, a(20) = 18384457, a(22) = 146482609, a(24) = 13882141, a(27) = 92672671, a(30) = 238997941, a(32) = 85276009, a(36) = 180467833. - Robert G. Wilson v, Mar 06 2018

			a(2) = 91 because 91 = 1^2 + 1*9 + 9^2 = 5^2 + 5*6 + 6^2 and 91 is the least number with this property.

Robert G. Wilson v, Solutions of a(n) for n <= 16

Cf. A024614, A118886, A198799, A327796, A327797.

nmx = 4750; t = Split@ Sort@ Flatten@ Table[x^2 + x*y + y^2, {x, nmx}, {y, x, nmx}]; lmt = 1 + Length@ t; f[n_] := Block[{k = 1}, While[Length@ t[[k]] != n && k < lmt, k++]; t[[k]][[1]]]; Array[f, 16] (* Robert G. Wilson v, Mar 06 2018 *)

N(n,d)=sum(x=1,sqrt(n\3),sum(y=max(x,sqrtint(n-x^2)\2),sqrtint(n-2*x^2),x^2+x*y+y^2==n&&!(d&&printf("%d",[x,y])))) \\ Set 2nd arg = 1 to display all decompositions.
a(n)=for(k=0,oo,N(k)==n&&return(k))

If A198799(n) is not a square and there is no square s < A198799(n) such that A088534(s) = n + 1, then a(n) = A198799(n), for all n > 0.

If A198799(n+1) is a square, then a(n) <= A198799(n+1).

a(17)-a(18) from Giovanni Resta, Mar 16 2018

a(19)-a(25) from Bert Dobbelaere, Feb 18 2023

A135412 Integers that equal three times the Heronian mean of two positive integers.

Original entry on oeis.org

3, 6, 7, 9, 12, 13, 14, 15, 18, 19, 21, 24, 26, 27, 28, 30, 31, 33, 35, 36, 37, 38, 39, 42, 43, 45, 48, 49, 51, 52, 54, 56, 57, 60, 61, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 81, 84, 86, 87, 90, 91, 93, 95, 96, 97, 98, 99, 102, 103, 104, 105, 108, 109, 111

Offset: 1

Pahikkala Jussi, Feb 17 2008

The Heronian mean of two nonnegative real numbers x and y is (x + y + sqrt(xy))/3. Therefore any number n is the Heronian mean of x = 3n and y = 0 (and also of x = n and y = n).

In particular, the sequence contains all numbers n = 3k which equal three times the Heronian mean of k and itself. If the two integers are required to be distinct then most multiples of 3 are no longer in the sequence: see A050931 for the sequence of integers that equal the Heronian mean of two distinct positive integers. Writing x = r^2*s where s is squarefree, the square root is an integer iff y = k^2*s for some integer k, and thus n = s*(r^2 + k^2 + rk). Therefore this sequence consists of the numbers listed in A024614 and their multiples by squarefree s. - M. F. Hasler, Aug 17 2016

			35 is in the sequence since 5 + 20 + sqrt(5*20) = 35.

Planet Math, Heronian mean is between geometric and arithmetic mean
Eric W. Weisstein, Heronian Mean. From MathWorld--A Wolfram Web Resource.
Wikipedia, Heronian mean

Cf. A024614, A050931, A004612.

Edited and definition corrected, following a remark by Robert Israel, by M. F. Hasler, Aug 17 2016

A336607 Numbers of the form x^3 + x^2y + xy^2 + y^3, where x and y are positive integers.

Original entry on oeis.org

4, 15, 32, 40, 65, 85, 108, 120, 156, 175, 203, 256, 259, 272, 320, 369, 400, 405, 477, 500, 520, 580, 585, 671, 680, 715, 803, 820, 864, 888, 935, 960, 1080, 1105, 1111, 1157, 1248, 1261, 1372, 1400, 1417, 1464, 1484, 1624, 1625, 1695, 1755, 1820, 1875, 1885

Offset: 1

César Eliud Lozada, Jul 27 2020

Numbers of the form (x+y)(x^2+y^2), where x and y are positive integers. - Chai Wah Wu, Aug 08 2020

No terms == 2 (mod 4). - Robert Israel, Sep 21 2020

			4=1^3+1^2*1+1*1^2+1^3, 15=1^3+1^2*2+1*2^2+2^3, 32=2^3+2^2*2+2*2^2+2^3, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A024614.

N:= 10000: # for terms <= N
S:= {}:
for x from 1 while (x+1)*(x^2+1) < N do
   V:= select(`<=`,map(y -> (x+y)*(x^2+y^2), {seq(i,i=1..min(x,(N-x^3)/x^2))}),N);
   S:= S union V;
od:
sort(convert(S,list)); # Robert Israel, Sep 21 2020

max = 5000; T0 = {}; xm = Ceiling[Sqrt[max]]; While[T = T0;
T0 = Table[x^3 + x^2 y + x y^2 + y^3, {x, 1, xm}, {y, x, xm}] //
     Flatten // Union // Select[#, # <= max &] &; T != T0, xm = 2 xm];
T

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A024615 Positions of even numbers in A024614.

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A024616 Positions of odd numbers in A024614.

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A024617 Position of n^2 + n + 1 in A024614.

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A024618 Positions of primes in A024614.

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A353198 Intersection of A000404 and A024614.

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A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

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A024606 Numbers of form x^2 + xy + y^2 with distinct x and y > 0.

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A300419 Smallest nonnegative number k such that k can be written in exactly n ways as x^2 + xy + y^2 where x and y are positive integers, with x >= y.

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A135412 Integers that equal three times the Heronian mean of two positive integers.

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A336607 Numbers of the form x^3 + x^2*y + x*y^2 + y^3, where x and y are positive integers.

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A336607 Numbers of the form x^3 + x^2y + xy^2 + y^3, where x and y are positive integers.