A084301 -id:A084301 - cp's OEIS frontend

A105826 a(n) = sigma(n) (mod 7).

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

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.

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).

A105826:= n-> (numtheory[sigma](n) mod 7):
seq (A105826(n), n=1..105); # Jani Melik, Jan 26 2011

Mod[DivisorSigma[1,Range[110]],7] (* Harvey P. Dale, Jul 30 2021 *)

PARI
```
a(n)=sigma(n)%7
```

A097011 Remainder of sigma(n) modulo 30.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 1, 18, 9, 20, 12, 2, 6, 24, 0, 1, 12, 10, 26, 0, 12, 2, 3, 18, 24, 18, 1, 8, 0, 26, 0, 12, 6, 14, 24, 18, 12, 18, 4, 27, 3, 12, 8, 24, 0, 12, 0, 20, 0, 0, 18, 2, 6, 14, 7, 24, 24, 8, 6, 6, 24, 12, 15, 14, 24, 4, 20, 6, 18, 20, 6, 1, 6

Offset: 1

Labos Elemer, Aug 19 2004

nonn

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A084301, A084303, A000203, A097012, A097013, A097014, A097015.

Table[Mod[DivisorSigma[1, n], 30], {n, 120}] (* Michael De Vlieger, Jan 10 2017 *)

a(n) = sigma(n) % 30; \\ Michel Marcus, Dec 19 2013

a(n) = mod(A000203(n), 30).

A097012 a(n) = sigma(n) mod 210.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96

Offset: 1

Labos Elemer, Aug 19 2004

nonn

Agrees with A000203(n) for n <= 89; sigma(90) = 234. - Omar E. Pol, Feb 02 2013

This sequence is not multiplicative. For example, at a(84) = 14 which is not a(3)*a(4)*a(7) = 224. - Andrew Howroyd, Aug 23 2018

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000203, A084301, A084303, A097011, A097013, A097014, A097015.

[SumOfDivisors(n) mod 210: n in [1..70]]; // Vincenzo Librandi, Jul 25 2017

Array[Mod[DivisorSigma[1, #], 210] &, 69] (* Michael De Vlieger, Jul 24 2017 *)

a(n) = sigma(n) % 210; \\ Michel Marcus, Dec 19 2013

a(n) = A000203(n) mod 210.

A097014 Smallest x such that sigma(x) mod 210 = n.

Original entry on oeis.org

104, 1, 211, 2, 3, 8311689, 5, 4, 7, 3698, 399, 130321, 6, 9, 13, 8, 1477, 2458624, 10, 725904, 19, 676, 6751, 18766224, 14, 52707600, 489, 24649, 12, 220433409, 29, 16, 21, 1250, 2779, 3694084, 22, 5184, 37, 18, 27, 1382976, 20, 729, 43, 128, 217

Offset: 0

Labos Elemer, Aug 19 2004

Compare with A084303 and A097013.

Row 210 of A074625 (apart from different ordering). - Michel Marcus, Dec 19 2013

			Several numbers belonging to odd residues are either large squares or twice-squares.
E.g.: r = n = 209: a(209) = 36324729 = 6027^2, sigma(6027^2) = 210*172974 + 209.
Full set is easily available running through squares or twice squares.

Michel Marcus, Table of n, a(n) for n = 0..209

Cf. A000203, A084301, A084303, A097011, A097012, A097013, A097015.

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

A097015 Smallest k such that sigma(k) + 1 is divisible by primorial(n).

Original entry on oeis.org

1, 1, 2401, 2614689, 36324729, 36324729, 2411675443849, 2411675443849, 12361036649679601

Offset: 0

Labos Elemer, Aug 19 2004

nonn

10^19 < a(9) <= 725298909352131113041. Terms a(3) through a(8) all have the prime signature p^4*q^2*r^2. Any x such that sigma(x) = -1 (mod 30) must have at least eight prime factors. However, for all n, there are solutions with fewer than three distinct prime factors. More generally, for any k > 1, let p be a prime of the form mk+1; then sigma(p^(k-2)) = -1 (mod k). For a(9), 725298909352131113041 is the least solution with eight prime factors. I have not been able to rule out a smaller solution with more prime factors. - David Wasserman, Dec 14 2007

Cf. A002110 (primorial), A084301, A084303, A000203, A097011, A097012, A097014, A233929.

a(n) = A233929(A002110(n)). - Andrew Howroyd, Dec 12 2024

More terms from David Wasserman, Dec 14 2007

a(0)=1 prepended by Andrew Howroyd, Dec 12 2024

A097013 Smallest x such that sigma(x) mod 30 = n.

Original entry on oeis.org

24, 1, 21, 2, 3, 923521, 5, 4, 7, 18, 27, 2401, 6, 9, 13, 8, 217, 9604, 10, 1089, 19, 98, 91, 21609, 14, 14641, 28, 49, 12, 2614689

Offset: 0

Labos Elemer, Aug 19 2004

Compare with A084303.

Row 30 of A074625 (apart from different ordering). - Michel Marcus, Dec 19 2013

Cf. A000203, A084301, A084303, A097011, A097012, A097014, A097015.

t=Table[Mod[DivisorSigma[1, w], 30], {w, 1, 2700000}]; Table[Min[Flatten[Position[t, j]]], {j, 0, 29}]

A105853 a(n) = sigma(n) (mod 10), i.e., unit's digit of sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 2, 8, 5, 3, 8, 2, 8, 4, 4, 4, 1, 8, 9, 0, 2, 2, 6, 4, 0, 1, 2, 0, 6, 0, 2, 2, 3, 8, 4, 8, 1, 8, 0, 6, 0, 2, 6, 4, 4, 8, 2, 8, 4, 7, 3, 2, 8, 4, 0, 2, 0, 0, 0, 0, 8, 2, 6, 4, 7, 4, 4, 8, 6, 6, 4, 2, 5, 4, 4, 4, 0, 6, 8, 0, 6, 1, 6, 4, 4, 8, 2, 0, 0, 0, 4, 2, 8, 8, 4, 0, 2, 8, 1, 6, 7, 2, 6, 4, 0, 2

Offset: 1

Shyam Sunder Gupta, May 05 2005

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A010879.

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), A105852 (k=9), A105853 (k=10).

A105853:=n->(numtheory)[sigma](n) mod 10: seq(A105853(n), n=1..180); # Wesley Ivan Hurt, Nov 07 2017

Mod[DivisorSigma[1,Range[110]],10] (* Harvey P. Dale, May 20 2019 *)

PARI
```
a(n)=sigma(n)%10
```

a(n) = A010879(A000203(n)). - Michel Marcus, Jul 26 2017

A084302 Remainder of tau(n) modulo 6.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 0, 2, 4, 4, 5, 2, 0, 2, 0, 4, 4, 2, 2, 3, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 3, 2, 4, 4, 2, 2, 2, 2, 0, 0, 4, 2, 4, 3, 0, 4, 0, 2, 2, 4, 2, 4, 4, 2, 0, 2, 4, 0, 1, 4, 2, 2, 0, 4, 2, 2, 0, 2, 4, 0, 0, 4, 2, 2, 4, 5, 4, 2, 0, 4, 4, 4, 2, 2, 0, 4, 0, 4, 4, 4, 0, 2, 0, 0, 3, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Jun 02 2003

The sums of the first 10^k terms, for k = 1, 2, ..., are 27, 236, 2275, 22166, 220070, 2195376, 21933228, 219259514, 2192385128, 21923168052, ... . Conjecture: the asymptotic mean of this sequence is 3*zeta(3)/zeta(2) = 3 * A253905 = 2.192288... . The conjecture is true if A211337 and A211338 have an equal asymptotic density (see also A059269). - Amiram Eldar, Jul 11 2024

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A054763, A084299, A084300, A084301.

Cf. A059269, A211337, A211338

Mod[DivisorSigma[0,Range[110]],6] (* Harvey P. Dale, Sep 04 2020 *)

A084302(n) = (numdiv(n)%6); \\ Antti Karttunen, Jul 07 2017

a(n) = A000005(n) modulo 6.

A097017 a(n) = sigma(5*n) modulo 6.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Labos Elemer, Aug 23 2004

nonn

a(n) = 0 if A112765(n) is even. - Robert Israel, Feb 27 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A084301, A112765.

f:= n -> numtheory:-sigma(5*n) mod 6;
map(f, [$1.100]); # Robert Israel, Feb 27 2018

Mod[#,6]&/@DivisorSigma[1,5*Range[110]] (* Harvey P. Dale, Jan 14 2019 *)

a(n) = sigma(5*n) % 6; \\ Michel Marcus, Mar 08 2014

a(n) = A084301(5*n).

Definition clarified by Harvey P. Dale, Jan 14 2019

cp's OEIS Frontend

A105826 a(n) = sigma(n) (mod 7).

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A097011 Remainder of sigma(n) modulo 30.

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A097012 a(n) = sigma(n) mod 210.

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A097014 Smallest x such that sigma(x) mod 210 = n.

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A074629 Duplicate of A067051.

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A097015 Smallest k such that sigma(k) + 1 is divisible by primorial(n).

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A097013 Smallest x such that sigma(x) mod 30 = n.

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A105853 a(n) = sigma(n) (mod 10), i.e., unit's digit of sigma(n).

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