A072461 -id:A072461 - 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

A072462 Numbers k such that sigma(k) == 3 (mod 4).

Original entry on oeis.org

2, 4, 8, 16, 18, 25, 32, 36, 64, 72, 98, 128, 144, 162, 169, 196, 225, 242, 256, 288, 289, 324, 392, 484, 512, 576, 648, 722, 784, 841, 882, 968, 1024, 1058, 1152, 1225, 1250, 1296, 1369, 1444, 1458, 1521, 1568, 1681, 1764, 1922, 1936, 2025, 2048, 2116, 2178

Offset: 1

Labos Elemer, Jun 19 2002

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A000203, A028982, A072461.

Select[Range[2200], Mod[DivisorSigma[1, #], 4] == 3 &] (* Michael De Vlieger, Aug 10 2023 *)

isok(n) = ((sigma(n) % 4) == 3); \\ Michel Marcus, Dec 19 2013

Name edited by Michel Marcus, Dec 19 2013

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, ...

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.

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

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A072462 Numbers k such that sigma(k) == 3 (mod 4).

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

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Mathematica

PARI

Formula