A055709 -id:A055709 - cp's OEIS frontend

A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 228, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6

Offset: 1

Labos Elemer, Dec 06 2001

nonn

a(n) <= 2p, where p = A002586(n) is the smallest prime factor of (1 + 2^n). (Proof. Since sigma_n(2p) = (1 + 2^n)(1 + p^n) and p is odd, 2p divides sigma_n(2p).) - Jonathan Sondow, Nov 23 2012

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A001158, A001159, A002586, A007691, A046762-A046764, A055709-A055717.

Cf. A218860, A218861 (unique values and where they first occur).

Table[m = 2; While[Mod[DivisorSigma[n, m], m] > 0, m++]; m, {n, 100}] (* T. D. Noe, Nov 23 2012 *)

Sum{d^n} = ka(n), d runs over the divisors of a(n), where k is an integer and a(n) is the smallest suitable number.

Definition and formulas corrected by Jonathan Sondow, Nov 23 2012

A055712 Numbers k such that k | sigma_8(k).

Original entry on oeis.org

1, 84, 156, 204, 364, 476, 514, 1092, 1428, 2316, 2652, 2892, 6069, 6188, 6748, 12138, 12532, 16212, 16388, 18564, 20244, 24276, 30108, 37596, 39372, 49164, 63291, 78897, 87724, 99202, 114716, 126582, 147679, 157794, 167331

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_8(k) is the sum of the 8th powers of the divisors of k (A013956).

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

Amiram Eldar, Table of n, a(n) for n = 1..1000
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A013956.

Cf. A055709, A055710, A055711, A055713.

Do[If[Mod[DivisorSigma[8, n], n]==0, Print[n]], {n, 1, 10000}]
Select[Range[170000],Divisible[DivisorSigma[8,#],#]&] (* Harvey P. Dale, Sep 15 2019 *)

is(n)=sigma(n,8)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A055710 Numbers k such that k | sigma_6(k).

Original entry on oeis.org

1, 10, 26, 60, 65, 130, 150, 228, 260, 442, 650, 780, 876, 988, 1105, 1140, 1460, 1690, 1950, 2210, 2850, 2964, 3211, 3796, 4380, 4420, 4940, 5070, 5475, 5548, 6010, 6422, 8840, 9633, 10950, 11050, 11388, 11972, 12350, 12818, 13260, 13756, 14820, 16644

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_6(k) is the sum of the 6th powers of the divisors of k (A013954).

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..450 from Harvey P. Dale)
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A013954.

Cf. A055709, A055711, A055712, A055713.

Do[If[Mod[DivisorSigma[6, n], n]==0, Print[n]], {n, 1, 25000}]
Select[Range[20000],Divisible[DivisorSigma[6,#],#]&] (* Harvey P. Dale, Jun 04 2015 *)

is(n)=sigma(n,6)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A055711 Numbers k such that k | sigma_7(k).

Original entry on oeis.org

1, 6, 28, 86, 120, 145, 258, 290, 435, 496, 580, 588, 672, 696, 870, 946, 1032, 1305, 1720, 1740, 2245, 2610, 2712, 2838, 3164, 3282, 3408, 3480, 3724, 3784, 4060, 4490, 5160, 5220, 6735, 6786, 6960, 7830, 8514, 8980, 9436, 9492, 9632, 9976

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_7(k) is the sum of the 7th powers of the divisors of k (A013955).

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A013955.

Cf. A055709, A055710, A055712, A055713.

Do[If[Mod[DivisorSigma[7, n], n]==0, Print[n]], {n, 1, 10000}]

is(n)=sigma(n,7)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A055713 Numbers k such that k | sigma_9(k).

Original entry on oeis.org

1, 6, 38, 42, 54, 114, 120, 135, 168, 190, 216, 222, 266, 270, 280, 285, 312, 342, 378, 456, 496, 540, 570, 672, 728, 760, 798, 840, 888, 945, 1026, 1064, 1080, 1140, 1330, 1512, 1554, 1560, 1710, 1782, 1806, 1862, 1890, 1962, 1976, 1995, 1998, 2160, 2166

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_9(k) is the sum of the 9th powers of the divisors of k (A013957).

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A013957.

Cf. A055709, A055710, A055711, A055712.

Do[If[Mod[DivisorSigma[9, n], n]==0, Print[n]], {n, 1, 5000}]

is(n)=sigma(n,9)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A066284 a(n) = A066135(4*n).

Original entry on oeis.org

34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 386, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194

Offset: 1

Labos Elemer, Dec 11 2001

nonn

a(n) <= 2p, where p = A002586(4n) is the least prime factor of (1 + 16^n). (See the Comment in A066135.) - Jonathan Sondow, Nov 23 2012

			First 3 terms correspond to entries of other sequences as follows: a(1)=A046763(2), a(2)=A055712(2), a(3)=A055716(2).
From _Michael De Vlieger_, Jul 17 2017: (Start)
First position of values, with observations pertaining to values for 1 <= n <= 3000:
    Value   Position   Observations:
    --------------------------------
       34     1        All odd.
       84     2        In A047235.
      194     6        In A017593.
      228    12
      386    36
     1282    72
     1538   144
     3084   288
   147468   576
     1956   864
  1046532  1152
    24578  2304
     3252  2880
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A066135, A046761, A046762, A046763, A055709-A055717, A007691, A001159.

Table[m = 2; While[Mod[DivisorSigma[4 n, m], m] > 0, m++]; m, {n, 66}] (* Michael De Vlieger, Jul 17 2017 *)

a(n) = {n *= 4; my(m = 2); while (sigma(m, n) % m, m++); m;} \\ Michel Marcus, Oct 02 2016

a(n) = Min{x : sigma_4n(x) mod x = 0, x > 1}

A076231 Numbers k such that sigma(k)/k, sigma_3(k)/k and sigma_5(k)/k are all integers.

Original entry on oeis.org

1, 6, 120, 672, 8128, 30240, 32760, 33550336, 459818240, 1379454720, 1476304896, 8589869056, 31998395520, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 30823866178560, 796928461056000, 6088728021160320, 14942123276641920, 212517062615531520

Offset: 1

Labos Elemer, Oct 03 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..86

Intersection of A007691, A046763 and A055709.

Cf. A000203, A001158, A001160, A007691, A046763, A055709, A076233, A076234.

isok(n) = !(sigma(n) % n) && !(sigma(n, 3) % n) && !(sigma(n, 5) % n); \\ Michel Marcus, Dec 26 2013

a(16)-a(20) from Donovan Johnson, May 08 2010

a(21)-a(23) from Amiram Eldar, May 09 2024

A076233 Numbers k such that sigma(k)/k and sigma_3(k)/k are both integers.

Original entry on oeis.org

1, 6, 120, 496, 672, 8128, 30240, 32760, 523776, 23569920, 33550336, 459818240, 1379454720, 1476304896, 8589869056, 31998395520, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 30823866178560, 796928461056000, 6088728021160320, 14942123276641920

Offset: 1

Labos Elemer, Oct 04 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..156

Cf. A000203, A001158, A007691, A046763, A055709, A076231, A076234.

isok(n) = !(sigma(n) % n) && !(sigma(n, 3) % n); \\ Michel Marcus, Dec 26 2013

a(19)-a(25) from Amiram Eldar, May 09 2024

A076234 Numbers k such that sigma(k)/k, sigma_3(k)/k, sigma_5(k)/k and sigma_7(k)/k are all integers.

Original entry on oeis.org

1, 6, 120, 672, 30240, 32760, 33550336, 459818240, 1379454720, 8589869056, 31998395520, 51001180160, 137438691328, 153003540480, 30823866178560, 796928461056000, 6088728021160320, 212517062615531520, 2305843008139952128, 69357059049509038080, 143573364313605309726720

Offset: 1

Labos Elemer, Oct 04 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..55

Cf. A000203, A001158, A001160, A013955.
Cf. A007691, A046763, A055709, A076231, A076233.
Cf. A066289 (k divides sigma_m(k) for all odd m).

isok(n) = !(sigma(n) % n) && !(sigma(n, 3) % n) && !(sigma(n, 5) % n) && !(sigma(n, 7) % n); \\ Michel Marcus, Dec 26 2013

a(13)-a(18) from Donovan Johnson, May 08 2010

a(19)-a(21) from Amiram Eldar, May 09 2024

A055715 Numbers k such that k | sigma_11(k).

Original entry on oeis.org

1, 6, 28, 120, 402, 496, 644, 672, 920, 1366, 1608, 1932, 2680, 2760, 3417, 3966, 4098, 4623, 4975, 5152, 6210, 6834, 8040, 8128, 8280, 9246, 9528, 9950, 12294, 13668, 15008, 15456, 15864, 16392, 18492, 19900, 24120, 24840, 25954, 27320, 27336, 29850, 30240, 32760

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_11(k) is the sum of the 11th powers of the divisors of k (A013959).

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

Cf. A013959.

Cf. A007691, A046762, A046763, A046764, A055709, A055710, A055711, A055712, A055713, A055714.

Do[If[Mod[DivisorSigma[11, n], n]==0, Print[n]], {n, 1, 40000}]

isok(k) = (sigma(k, 11) % k) == 0; \\ Michel Marcus, Nov 09 2019

a(37)-a(40) corrected and more terms added by Amiram Eldar, Nov 09 2019

cp's OEIS Frontend

A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

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A055712 Numbers k such that k | sigma_8(k).

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A055710 Numbers k such that k | sigma_6(k).

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A055711 Numbers k such that k | sigma_7(k).

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A055713 Numbers k such that k | sigma_9(k).

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A066284 a(n) = A066135(4*n).

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A076231 Numbers k such that sigma(k)/k, sigma_3(k)/k and sigma_5(k)/k are all integers.

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A076233 Numbers k such that sigma(k)/k and sigma_3(k)/k are both integers.

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