A072862 -id:A072862 - cp's OEIS frontend

A074384 Solutions to mod(sigma(x), 6) = 5.

2401, 9604, 21609, 28561, 38416, 60025, 86436, 114244, 130321, 153664, 194481, 240100, 257049, 290521, 345744, 456976, 521284, 540225, 614656, 693889, 714025, 777924, 923521, 960400, 1028196, 1162084, 1172889, 1270129, 1382976, 1500625

Offset: 1

Labos Elemer, Aug 22 2002

nonn

			4th powers of primes of the form 6k+1 are here because sigma[p^4]=p^4+p^3+p^2+p+1 congruent 1+1+1+1+1=5 mod 6. There are also other fourth powers, like 38416=(2*7)^4, 194481=(3*7)^4, 456976=(2*13)^4, and solutions which are not fourth powers like 9604=2^2*7^4 and 21609=3^2*7^4.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203, A072862, A072461, A072462.

Do[s=Mod[DivisorSigma[1, n], 6]; If[s==5, Print[n]], {n, 1, 1000000}]
Select[Range[1600000],Mod[DivisorSigma[1,#],6]==5&] (* Harvey P. Dale, Jul 06 2014 *)

{n: A084301(n) = 5}. - R. J. Mathar, May 19 2020

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

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, ...

A049605 Smallest k>1 such that k divides sigma(k*n).

Original entry on oeis.org

6, 3, 2, 6, 2, 2, 2, 3, 6, 2, 2, 2, 2, 2, 2, 6, 2, 3, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 6, 2, 2, 2, 2, 2

Offset: 1

Benoit Cloitre, Jul 26 2002

a(n) = 2, 3 or 6. For any m, a(A028983(m)) = 2. If a(m)=6 then m is a square but if m is a square a(m) is not necessarily 6, first example is 7: a(7^2)=3 (cf. A072864).

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A028983 (locations of 2), A067051 (locations of 3), A072862 (locations of 6).

A049605 := proc(n)
    for k from 2 do
        if modp(numtheory[sigma](k*n),k) = 0 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Oct 26 2015

sk[n_]:=Module[{k=2},While[!Divisible[DivisorSigma[1,k*n],k],k++];k]; sk /@ Range[110] (* Harvey P. Dale, Jan 04 2015 *)

a(n) = {k = 2; while(sigma(k*n) % k, k++); k ;} \\ Michel Marcus, Nov 21 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

cp's OEIS Frontend

A074384 Solutions to mod(sigma(x), 6) = 5.

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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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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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A049605 Smallest k>1 such that k divides sigma(k*n).

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A074629 Duplicate of A067051.

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