A074384 -id:A074384 - 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

A074628 Numbers k such that sigma(k) == 2 mod 6.

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

Offset: 1

Labos Elemer, Aug 26 2002

			For k=39: sigma(39) = 1+3+13+39 = 56 = 6*9 + 2.

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

Cf. A000203, A072862, A074384, A074627, A074628, A074629, A074630.

Select[Range[300],Mod[DivisorSigma[1,#],6]==2&] (* Harvey P. Dale, Nov 14 2014 *)

isok(n) = ((sigma(n) % 6) == 2); \\ Michel Marcus, Dec 19 2013

A000203(n) mod 6 = 2.

{n: A084301(n) = 2}. - R. J. Mathar, May 19 2020

A072862 The smallest k>1 such that k divides sigma(k*n) is equal to 6.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 225, 256, 289, 324, 400, 484, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761, 5041

Offset: 1

Benoit Cloitre, Jul 26 2002

The smallest k>1 such that k divides sigma(k*n) is always 2, 3 or 6. Sequence generates squares but not all the squares (49, 169, 196, 361, 441, 676, 784, ... are the first squares not in the sequence).

This is not the sequence of solutions to sigma(n) == 1 (mod 6), because 2401 is in this sequence but sigma(2401) == 5 (mod 6). See A074384 for more examples. - R. J. Mathar, May 19 2020

{n: A049605(n) = 6}. - R. J. Mathar, May 19 2020

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

A074625 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[sigma[x],n]=k.

Original entry on oeis.org

1, 1, 3, 1, 7, 2, 1, 5, 2, 3, 1, 4, 2, 3, 8, 1, 7, 2, 3, 2401, 5, 1, 29, 2, 3, 6, 5, 4, 1, 10, 2, 3, 9, 5, 4, 7, 1, 19, 2, 3, 13, 5, 4, 7, 10, 1, 6, 2, 3, 8, 5, 4, 7, 18, 19, 1, 9, 2, 3, 24, 5, 4, 7, 16, 21, 43, 1, 13, 2, 3, 2401, 5, 4, 7, 49, 31213, 9604, 6, 1, 8, 2, 3, 10, 5, 4, 7, 33, 22

Offset: 1

Labos Elemer, Aug 26 2002

In the table output, one can observe constant diagonals (or lines in the square output). The indices of these are: 1, 3, 4, 6, 7, 8, 12, 13, ... (see A002191). And the corresponding values are: 1, 2, 3, 5, 4, 7, 6, 9, ... (see A002192). - Michel Marcus, Dec 19 2013

			Triangle begins
1;
1,3;
1,7,2;
1,5,2,3;
1,4,2,3,8; ...

Cf. A000203, A028982, A074384, A074386, A072461, A072462, A072862.

{k=0, s=0, fl=1}; Table[Print["#"]; Table[fl=1; Print[{r, m}]; Do[s=Mod[DivisorSigma[1, n], m]; If[(s==r)&&(fl==1), Print[n]; fl=0], {n, 1, 150000}], {r, 0, m-1}], {m, 1, 25}]

Min{x; Mod[sigma[x], n]=r}, r=1..n, n=1, ...

A084303 Smallest x such that sigma(x) mod 6 = n.

Original entry on oeis.org

5, 1, 7, 2, 3, 2401

Offset: 0

Labos Elemer, Jun 02 2003

Row 6 of A074625 (apart from different ordering). - Michel Marcus, Dec 19 2013

			n=5: sigma(2401) = 1+7+49+343+2401 = 2801 = 6*466+5, hence a(5)=2401.

Cf. A000203, A074384.

Table[k = 1; While[Mod[DivisorSigma[1, k], 6] != n, k++]; k, {n, 0, 5}] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = {my(x = 1); while((sigma(x) % 6) != n, x++); x;} \\ Michel Marcus, Dec 18 2013

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

A331121 a(n) is the smallest positive integer k for which tau(k) does not divide sigma(n).

Original entry on oeis.org

2, 2, 4, 2, 6, 16, 4, 2, 2, 6, 16, 4, 4, 16, 16, 2, 6, 2, 4, 6, 4, 16, 16, 24, 2, 6, 4, 4, 6, 16, 4, 2, 16, 6, 16, 2, 4, 24, 4, 6, 6, 16, 4, 16, 6, 16, 16, 4, 2, 2, 16, 4, 6, 36, 16, 36, 4, 6, 24, 16, 4, 16, 4, 2, 16, 16

Offset: 1

Lechoslaw Ratajczak, Jan 10 2020

nonn

Consecutive t satisfying the equation a(t) = 2 are consecutive elements of A028982 (squares and twice squares).

Conjecture: consecutive u satisfying the equation a(u) = 4 are consecutive elements of a sequence defined as follows: (A024614 \ A088535) \ A074384. The conjecture was checked for 10^6 consecutive integers.

			a(10) = 6 because sigma(10) = 18 is divisible by (tau(1) = 1), (tau(2) = 2), (tau(3) = 2), (tau(4) = 3), (tau(5) = 2), and is not divisible by (tau(6) = 4).

Cf. A000005, A000203, A024614, A028982, A074384, A088535.

Array[Block[{k = 1}, While[Mod[DivisorSigma[1, #], DivisorSigma[0, k]] == 0, k++]; k] &, 66] (* Michael De Vlieger, Jan 31 2020 *)

a(n):=(for k:1 while mod(divsum(n), length(divisors(k))) = 0 do z:k, z+1) $ makelist(a(n), n, 1, 100, 1);

a(n) = my(k=1, sn=sigma(n)); while ((sn % numdiv(k)) == 0, k++); k; \\ Michel Marcus, Jan 10 2020

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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A074628 Numbers k such that sigma(k) == 2 mod 6.

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A072862 The smallest k>1 such that k divides sigma(k*n) is equal to 6.

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

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A074625 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[sigma[x],n]=k.

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A084303 Smallest x such that sigma(x) mod 6 = n.

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