A072462 -id:A072462 - 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

A105824 a(n) = sigma(n) mod 4.

Original entry on oeis.org

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

Offset: 1

Shyam Sunder Gupta, May 05 2005

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

Cf. A000203, A010873, A248150, A072461, A072462, A191217.

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).

A105824:= n-> (numtheory[sigma](n) mod 4):
seq (A105824(n), n=1..105); # Jani Melik, Jan 26 2011

Table[Mod[DivisorSigma[1, n], 4], {n, 100}] (* Wesley Ivan Hurt, Nov 07 2017 *)

PARI
```
a(n)=sigma(n)%4
```

a(n) = A010873(A000203(n)). - Antti Karttunen, Nov 07 2017

A072461 Numbers k such that sigma(k) == 1 (mod 4).

Original entry on oeis.org

1, 9, 49, 50, 81, 100, 121, 200, 338, 361, 400, 441, 450, 529, 578, 625, 676, 729, 800, 900, 961, 1089, 1156, 1352, 1600, 1682, 1800, 1849, 2209, 2312, 2401, 2450, 2704, 2738, 3042, 3200, 3249, 3362, 3364, 3481, 3600, 3969, 4050, 4225, 4489, 4624, 4761

Offset: 1

Labos Elemer, Jun 19 2002

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A000203, A028982, A072462, A081339.

Positions of 1's in A105824.

Select[Range[5000],Mod[DivisorSigma[1,#],4]==1&] (* Harvey P. Dale, Oct 24 2016 *)

PARI
```
isok(n) = ((sigma(n) % 4) == 1);
```

Name edited by Michel Marcus, Dec 19 2013 and Amiram Eldar, Jun 26 2024

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, ...

A324899 Odd numbers k for which sigma(k) == 3 (mod 4).

Original entry on oeis.org

25, 169, 225, 289, 841, 1225, 1369, 1521, 1681, 2025, 2601, 2809, 3025, 3721, 5329, 7569, 7921, 8281, 9025, 9409, 10201, 11025, 11881, 12321, 12769, 13225, 13689, 14161, 15129, 15625, 18225, 18769, 20449, 22201, 23409, 24025, 24649, 25281, 27225, 29929, 32761, 33489, 34969, 37249, 38809, 41209, 46225, 47961, 52441, 54289, 55225

Offset: 1

Antti Karttunen, Apr 20 2019

nonn

These are all squares. Square roots are in A324909.

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

Intersection of A016754 and A072462.
Cf. A000203, A324909 (square roots).
Cf. A325311 (a subsequence).

Select[Range[1, 235, 2]^2, Mod[DivisorSigma[1, #], 4] == 3 &] (* Amiram Eldar, Apr 05 2024 *)

isA324899(n) = ((n%2)&&3==(sigma(n)%4));

a(n) = A324909(n)^2. - Amiram Eldar, Apr 05 2024

A378998 Number of trailing 1-bits in the binary representation of sigma(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Dec 16 2024

nonn

Cf. A000203, A007814, A088580, A028982 (positions of terms > 0), A028983 (of 0's), A072461 (of 1's), A072462 (of terms > 1), A337195, A378999 [= a(n^2)].

IntegerExponent[DivisorSigma[1, Range[100]] + 1, 2] (* Paolo Xausa, Dec 19 2024 *)

PARI
```
A378998(n) = valuation(sigma(n)+1,2);
```

a(n) = A007814(A088580(n)). [the 2-adic valuation of 1+sigma(n)]

For all n in A028982, a(n) = A337195(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A105824 a(n) = sigma(n) mod 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A072461 Numbers k such that sigma(k) == 1 (mod 4).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A324899 Odd numbers k for which sigma(k) == 3 (mod 4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A378998 Number of trailing 1-bits in the binary representation of sigma(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula