A074628 -id:A074628 - cp's OEIS frontend

A084301 a(n) = sigma(n) mod 6.

1, 3, 4, 1, 0, 0, 2, 3, 1, 0, 0, 4, 2, 0, 0, 1, 0, 3, 2, 0, 2, 0, 0, 0, 1, 0, 4, 2, 0, 0, 2, 3, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 3, 3, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 4, 2, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 3, 0, 1, 0, 0, 2, 0, 0

Offset: 1

Labos Elemer, Jun 02 2003

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A000203, A010875, A054763, A084299, A084300.
Sequences sigma(n) mod k: A053866 (k=2), A074941 (k=3), A105824 (k=4), A105825 (k=5), A084301 (k=6), A105826 (k=7), A105827 (k=8).
Cf. A074627 (locations of 0), A074628 (locations of 2), A067051 (locations of 3), A074630 (locations of 4), A074384 (locations of 5).

A084301:= n-> (numtheory[sigma](n) mod 6):
seq (A084301(n), n=1..105); # Jani Melik, Jan 26 2011

Table[Mod[DivisorSigma[1,n],6],{n,120}]  (* Harvey P. Dale, Apr 07 2011 *)

a(n)=sigma(n)%6 \\ Charles R Greathouse IV, Mar 09 2014

a(n) = A010875(A000203(n)). - Antti Karttunen, Nov 07 2017

A067051 The smallest k>1 such that k divides sigma(k*n) is equal to 3.

Original entry on oeis.org

2, 8, 18, 32, 49, 50, 72, 98, 128, 162, 169, 196, 200, 242, 288, 338, 361, 392, 441, 450, 512, 578, 648, 676, 722, 784, 800, 882, 961, 968, 1058, 1152, 1225, 1250, 1352, 1369, 1444, 1458, 1521, 1568, 1682, 1764, 1800, 1849, 1922, 2048, 2178, 2312, 2450, 2592

Offset: 1

Benoit Cloitre, Jul 26 2002

The smallest m>1 such that m divides sigma(m*n) is 2, 3 or 6.
Appears to be the same sequence as A074629. - Ralf Stephan, Aug 18 2004. [Proof: Mathar link]
Square terms are in A074216. Nonsquare terms appear to be A001105 except {0}. - Michel Marcus, Dec 26 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. J. Mathar, OEIS A074629

Subsequence of A087943.

Cf. A072862, A074384, A074627, A074628, A074630.

[n: n in [1..3*10^3] | (SumOfDivisors(n) mod 6) eq 3]; // Vincenzo Librandi, Dec 11 2015

select(t -> numtheory:-sigma(t) mod 6 = 3, [$1..10000]); # Robert Israel, Dec 11 2015

Select[Range@ 2600, Mod[DivisorSigma[1, #], 6] == 3 &] (* Michael De Vlieger, Dec 10 2015 *)

isok(n) = (sigma(2*n) % 2) && !(sigma(3*n) % 3); \\ Michel Marcus, Dec 26 2013

{n: A000203(n) mod 6 = 3.}  (Old definition of A074629) - Labos Elemer, Aug 26 2002
In the prime factorization of n, no odd prime has odd exponent, and 2 has odd exponent or at least one prime == 1 (mod 6) has exponent == 2 (mod 6). - Robert Israel, Dec 11 2015
{n: A049605(n) = 3}. - R. J. Mathar, May 19 2020
{n: A084301(n) = 3 }. - R. J. Mathar, May 19 2020
A087943 INTERSECT A028982. - R. J. Mathar, May 30 2020

A074627 Numbers n such that sigma(n) is divisible by 6.

Original entry on oeis.org

5, 6, 10, 11, 14, 15, 17, 20, 22, 23, 24, 26, 29, 30, 33, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 51, 53, 54, 55, 56, 58, 59, 60, 62, 65, 66, 68, 69, 70, 71, 74, 77, 78, 80, 82, 83, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 99, 101, 102, 104, 105, 106, 107, 110, 113, 114, 115

Offset: 1

Labos Elemer, Aug 26 2002

n=10: sigma(10) = 1+2+5+10 = 18 = 3*6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000203, A072862, A074384, A074628, A067051, A074630.

Select[Range@ 120, Divisible[DivisorSigma[1, #], 6] &] (* Michael De Vlieger, Feb 25 2017 *)

isok(n) = !(sigma(n) % 6); \\ Michel Marcus, Dec 17 2013

A000203(n) modulo 6 = 0.
{n: A084301(n) = 0 }. - R. J. Mathar, May 19 2020
A087943 INTERSECT A028983. - R. J. Mathar, May 19 2020

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

A074630 Numbers k such that sigma(k) == 4 mod 6.

Original entry on oeis.org

3, 12, 27, 48, 75, 91, 108, 133, 192, 217, 243, 247, 259, 273, 300, 301, 343, 363, 364, 399, 403, 427, 432, 469, 481, 511, 532, 553, 559, 589, 651, 675, 679, 703, 721, 741, 763, 768, 777, 793, 817, 819, 867, 868, 871, 889, 903, 949, 972, 973, 988, 1027, 1029

Offset: 1

Labos Elemer, Aug 26 2002

			k=48 is a term because sigma(48) = 1+2+3+4+6+8+12+16+24+48 = 124 = 6*20 + 4. [corrected by _Harvey P. Dale_, Jan 17 2013]

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

Cf. A000203, A067051, A072862, A074384, A074627, A074628.

Select[Range[1100],Mod[DivisorSigma[1,#],6]==4&] (* Harvey P. Dale, Jan 17 2013 *)

Mod(A000203(n), 6) = 4.

{n: A084301(n) = 4}. - R. J. Mathar, May 19 2020

A074629 Duplicate of A067051.

Original entry on oeis.org

2, 8, 18, 32, 49, 50, 72, 98, 128, 162, 169, 196, 200, 242, 288, 338, 361, 392, 441, 450, 512, 578, 648, 676, 722, 784, 800, 882, 961, 968, 1058, 1152, 1225, 1250, 1352, 1369, 1444, 1458, 1521, 1568, 1682, 1764, 1800, 1849, 1922, 2048, 2178, 2312, 2450, 2592

Offset: 1

Labos Elemer, Aug 26 2002

dead

Square terms are in A074216. Nonsquare terms appear to be A001105 except {0}. - Michel Marcus, Dec 26 2013

In the prime factorization of n, no odd prime has odd exponent, and 2 has odd exponent or at least one prime == 1 (mod 6) has exponent == 2 (mod 6). - Robert Israel, Dec 11 2015

			n=32: sigma(32) = 63 = 6*10 + 3.

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

Cf. A000203, A072862, A074384, A074627, A074628, A074630.

Appears to be the same sequence as A067051. - Ralf Stephan, Aug 18 2004

[n: n in [1..3*10^3] | (SumOfDivisors(n) mod 6) eq 3]; // Vincenzo Librandi, Dec 11 2015

select(t -> numtheory:-sigma(t) mod 6 = 3, [$1..10000]); # Robert Israel, Dec 11 2015

Select[Range@ 2600, Mod[DivisorSigma[1, #], 6] == 3 &] (* Michael De Vlieger, Dec 10 2015 *)

isok(n) = (sigma(n) % 6) == 3; \\ Michel Marcus, Dec 26 2013

A000203(n) mod 6 = 3.

{n: A084301(n) = 3 }. - R. J. Mathar, May 19 2020

A097022 a(n) = (sigma(2n^2)-3)/6.

Original entry on oeis.org

0, 2, 6, 10, 15, 32, 28, 42, 60, 77, 66, 136, 91, 142, 201, 170, 153, 302, 190, 325, 370, 332, 276, 552, 390, 457, 546, 598, 435, 1007, 496, 682, 864, 767, 883, 1270, 703, 952, 1189, 1317, 861, 1852, 946, 1396, 1875, 1382, 1128, 2216, 1400, 1952, 1995, 1921

Offset: 1

Labos Elemer, Aug 24 2004

nonn

See A065764, A065765 and A097022.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A065764, A065765, A000203, A001105, A002117, A072682, A074627, A074628, A074629, A074630, A067051.

Table[(DivisorSigma[1,2n^2]-3)/6,{n,60}] (* Harvey P. Dale, Sep 12 2022 *)

a(n) = (sigma(2*n^2) - 3)/6; \\ Michel Marcus, Dec 20 2013

a(n) = (A065765(n)-3)/6 = A000203(A001105(n) - 3)/6.

Sum_{k=1..n} a(k) ~ c * n^3, where c = 4*zeta(3)/Pi^2 = 0.243587... . - Amiram Eldar, Oct 28 2022

A232436 Numbers which are uniquely decomposable into x^2+xy+y^2, the unique decomposition being with two distinct nonzero x and y.

Original entry on oeis.org

7, 13, 19, 21, 28, 31, 37, 39, 43, 52, 57, 61, 63, 67, 73, 76, 79, 84, 93, 97, 103, 109, 111, 112, 117, 124, 127, 129, 139, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 193, 199, 201, 208, 211, 219, 223, 228, 229, 237, 241, 244, 252, 268

Offset: 1

Jean-Christophe Hervé, Nov 23 2013

nonn

These are the primitive elements of A024606, the integers which are expressible as x^2 + xy + y^2 with distinct nonzero x and y.
As a subsequence of A003136 (Loeschian numbers), the sequence is related with the triangular lattice: circles with radius sqrt(a(n)) centered at a grid point in this lattice hit exactly 12 points, cf. A004016.
Numbers with exactly one prime factor of form 6k+1 with multiplicity one and no prime factor of form 3k+2 with odd multiplicity, that is a(n) is of form 3^a*p*q^2, with a>=0, p a prime of form 6k+1, and q an integer with all its prime factors of form 3k+2. There is thus no square in the sequence.
From a(n) = 3^a*p*q^2, it is easily seen that sigma(a(n)) = 2 mod 6,
thus this sequence is a subsequence of A074628: the two sequences are equal up to a(308) = 1723; then A074628(309)= 1729 = a(1)*a(2)*a(3),  the famous Ramanujan's taxi number, and a(309) = A074628(310) = 1731.
The square of these numbers is also uniquely decomposable into the form x^2 + xy + y^2 with x and y > 0, thus this sequence is a subsequence of A232437.

			a(1)= 7 = 2^2+2+1, a(2)= 13 = 3^2+3+1. However 3 = 1+1+1 and 4 = 2^2+0*2+0 are not in the sequence because the unique decomposition of these numbers is not with two distinct nonzero numbers; 49, 147 are also excluded because there are two decompositions of these numbers (including one with equal or zero components x and y).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 6364 terms from Jean-Christophe Hervé)
A. Mazel, I. Stuhl, Y. Suhov, Hard-core configurations on a triangular lattice and Eisenstein primes, arXiv:1803.04041 [math.PR], 2018.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003136, A024606, A002476, A232437.
Cf. A004016, A034020 (A004016 = 0), A230781 (A004016 = 6).
Cf. (Analog for the square lattice) A230779, A001481, A004431, A002144, A004018, A084645.

r[k_] := Reduce[x != 0 && y != 0 && x != y && k == x^2 + x y + y^2, {x, y}, Integers];
selQ[k_] := If[IntegerQ[Sqrt[k]], False, Which[rk = r[k]; rk === False, False, rk[[0]] === And && Length[rk] == 2, True, rk[[0]] === Or && Length[rk] == 12, True, True, False]];
Select[Range[1000], selQ] (* Jean-François Alcover, Feb 20 2020 *)

Terms are obtained by the products A230781(k)*A002476(p) for k, p > 0, ordered by increasing values.

A004016(a(n))=12.

cp's OEIS Frontend

A084301 a(n) = sigma(n) mod 6.

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A067051 The smallest k>1 such that k divides sigma(k*n) is equal to 3.

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A074627 Numbers n such that sigma(n) is divisible by 6.

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

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A074630 Numbers k such that sigma(k) == 4 mod 6.

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A074629 Duplicate of A067051.

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A097022 a(n) = (sigma(2n^2)-3)/6.

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