A067051 -id:A067051 - cp's OEIS frontend

A074629 Duplicate of A067051.

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

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

Original entry on oeis.org

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

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

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

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

A074216 Squares satisfying sigma(n)==0 (mod 3).

Original entry on oeis.org

49, 169, 196, 361, 441, 676, 784, 961, 1225, 1369, 1444, 1521, 1764, 1849, 2704, 3136, 3249, 3721, 3844, 3969, 4225, 4489, 4900, 5329, 5476, 5776, 5929, 6084, 6241, 7056, 7396, 8281, 8649, 9025, 9409, 10609, 10816, 11025, 11881, 12321, 12544

Offset: 1

Benoit Cloitre, Sep 17 2002

nonn

Seems to contain all numbers of form k^2*p^2 where p are primes in A002476, k is not congruent to p and >=1.

Squares in A067051. - Michel Marcus, Dec 26 2013

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

Cf. A067051, A065764.

[n: n in [1..14161]|IsSquare(n) and DivisorSigma(1,n) mod 3 eq 0 ]; // Marius A. Burtea, Aug 17 2019

with(numtheory); A074216:=n->`if`(1-ceil(sigma(n^2)/3)+floor(sigma(n^2)/3)=1,n^2,NULL); seq(A074216(n), n=1..200); # Wesley Ivan Hurt, Dec 06 2013

Select[Range[150]^2,Divisible[DivisorSigma[1,#],3]&] (* Harvey P. Dale, Jul 10 2012 *)

isok(n) = issquare(n) && !(sigma(n) % 3); \\ Michel Marcus, Aug 17 2019

Conjecture: a(n) = A072864(n)^2. - 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

A332217 Numbers k for which the 2-adic valuation of sigma(k) is zero and its 3-adic valuation is 1 (so that sigma(k) is an odd multiple of 3, but not of 9).

Original entry on oeis.org

2, 8, 18, 49, 50, 72, 128, 162, 169, 196, 200, 242, 361, 441, 450, 512, 578, 648, 676, 784, 961, 968, 1058, 1152, 1225, 1250, 1369, 1444, 1458, 1521, 1682, 1764, 1800, 1849, 2178, 2312, 2704, 3136, 3200, 3249, 3362, 3721, 3844, 3969, 4050, 4225, 4232, 4418, 4489, 4608, 4802, 4900, 5000

Offset: 1

Antti Karttunen, Feb 11 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Subsequence of A067051, which is a subsequence of A028982.
Cf. A332218 (a subsequence).
Cf. A000203, A007949, A332216.

Select[Range[5*10^3], IntegerExponent[DivisorSigma[1, #], {2, 3}] === {0, 1} &] (* Michael De Vlieger, Feb 12 2020 *)

isA332217(n) = ((sigma(n)%2)&&(valuation(sigma(n),3)==1));

{n: A000035(A000203(n))*A007949(A000203(n))==1}.

A332218 Numbers k such that A332221(k) = A156552(sigma(k)) is 2*{an odd square}.

Original entry on oeis.org

2, 162, 441, 2704, 4225, 275194921

Offset: 1

Antti Karttunen, Feb 11 2020

Any even term of A332216 must occur also in this sequence.

			  a(n)              -> sigma(a(n))              -> A156552(sigma(a(n)))
     2 = 2^1 * 1^2  ->    3 = 3^1               ->      2 = 2^1 * 1^1,
   162 = 2^1 * 3^4  ->  363 = 3^1 * 11^2        ->     98 = 2^1 * 7^2,
   441 = 3^2 * 7^2  ->  741 = 3^1 * 13^1 * 19^1 ->    578 = 2^1 * 17^2,
  2704 = 2^4 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,
  4225 = 5^2 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,
and
275194921 = 53^2 * 313^2 -> 281384229 = 3^1 * 7^1 * 181^2 * 409^1 -> 9671406556943421676716050 = 2^1 * 5^2 * 7^2 * 62829235873^2.

Index entries for sequences related to sigma(n)

Cf. A000203, A156552, A332216, A332217, A332221.

Subsequence of A332217 ⊂ A067051 ⊂ A028982.

Select[Range@ 5000, And[IntegerQ[#], OddQ[#]] &@ Sqrt[#/2] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ DivisorSigma[1, #]]] &] (* Michael De Vlieger, Feb 12 2020 *)

\\ Needs also code from A156552:
istosq(n) = ((1==valuation(n,2))&&issquare(n/2));
for(n=1,2^25,if(istosq(A156552(sigma(n*n))),print1(n*n,", ")); if(istosq(A156552(sigma(2*n*n))),print1(2*n*n,", ")));

cp's OEIS Frontend

A074629 Duplicate of A067051.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A074216 Squares satisfying sigma(n)==0 (mod 3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica