A109974 -id:A109974 - cp's OEIS frontend

A123926 Greatest common divisor of sigma_k(n) for all k >= 1.

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 4, 1, 2, 1, 2, 6, 4, 2, 2, 2, 1, 2, 4, 2, 2, 4, 2, 3, 4, 2, 4, 1, 2, 2, 4, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 12, 2, 2, 2, 1, 4, 4, 2, 6, 4, 4, 2, 1, 2, 2, 2, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 4, 2, 2, 2, 4, 6, 4, 2, 4, 6, 2, 3, 2, 1, 2, 4, 2, 2, 8

Offset: 1

Franklin T. Adams-Watters, Nov 21 2006

Has the property that if gcd(n,m) = 1, then a(n)*a(m) divides a(n*m). First inequality is a(4) = 1, a(5) = 2, but a(20) = 6. It appears that a(n) also always divides sigma_0(n) = tau(n).
Contribution from Matthew Vandermast, Feb 10 2010: (Start)
1. If an integer m does not divide sigma_0(n), m will also not divide sigma_(totient m)(n). Therefore a(n) always divides sigma_0(n) = tau(n).
2. a(n) is even iff sigma_1(n) is even. Cf. A028982, A028983.
3. a(p)=2 for any odd prime p. If n is an odd integer with 2^e divisors, then a(n)=2^e.
4. For any prime p and positive integer m, if p is congruent to 1 mod m, then a(p^(m-1))=m. It follows from Dirichlet's Theorem (see link) that every positive integer appears in the sequence infinitely often. (End)

			For n=4, sigma_1(n) = 7, sigma_2(n) = 21, both divisible by 7, but sigma_3(n) = 73, which is not, so a(4) = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Dirichlet's Theorem

Cf. A109974, A000005, A000203, A001157.

a[n_] :=  GCD @@ Table[DivisorSigma[k, n] , {k, 0, EulerPhi[n]}]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 21 2012 *)

a(n)=my(d=divisors(n), g=#d); for(k=1, eulerphi(n), g=gcd(lift(sum(i=1,#d,Mod(d[i],g)^k)),g); if(g<3,return(g))); g \\ Charles R Greathouse IV, Jun 17 2013

A319278 Square array sigma_k(n) read down antidiagonals: sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 7, 1, 17, 28, 21, 6, 1, 33, 82, 73, 26, 12, 1, 65, 244, 273, 126, 50, 8, 1, 129, 730, 1057, 626, 252, 50, 15, 1, 257, 2188, 4161, 3126, 1394, 344, 85, 13, 1, 513, 6562, 16513, 15626, 8052, 2402, 585, 91, 18, 1, 1025, 19684, 65793, 78126, 47450, 16808, 4369, 757, 130, 12

Offset: 1

R. J. Mathar, Sep 16 2018

Equals the square array A082771 without its first column.

			The array starts in row n=1 with columns k>=1 as:
     1      1      1      1      1      1       1        1
     3      5      9     17     33     65     129      257
     4     10     28     82    244    730    2188     6562
     7     21     73    273   1057   4161   16513    65793
     6     26    126    626   3126  15626   78126   390626
    12     50    252   1394   8052  47450  282252  1686434
     8     50    344   2402  16808 117650  823544  5764802
    15     85    585   4369  33825 266305 2113665 16843009

Columns: A000203, A001157, A001158, A001159.
Rows: A000012, A000051, A034472, A001576, A034474, A034488.
Cf. A082771, A023887 (diagonal), A109974, A319194 (partial column sums).

T[n_, k_] := DivisorSigma[k, n];
Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

sigma_k(n) = sum_{d|n} d^k.

A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

Original entry on oeis.org

1, 1, 5, 1, 7, 7, 1, 11, 13, 17, 1, 19, 31, 35, 11, 1, 35, 85, 95, 31, 35, 1, 67, 247, 311, 131, 91, 15, 1, 131, 733, 1127, 631, 341, 57, 49, 1, 259, 2191, 4295, 3131, 1615, 351, 155, 34, 1, 515, 6565, 16775, 15631, 8645, 2409, 775, 130, 55, 1, 1027, 19687, 66311, 78131, 49111, 16815, 4991, 850, 217, 23

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
   1,   1,    1,     1,     1,      1,  ...
   5,   7,   11,    19,    35,     67,  ...
   7,  13,   31,    85,   247,    733,  ...
  17,  35,   95,   311,  1127,   4295,  ...
  11,  31,  131,   631,  3131,  15631,  ...
  35,  91,  341,  1615,  8645,  49111,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Columns k=0..3 give A060640, A001001, A027847, A027848.

Cf. A109974, A320940 (diagonal), A321876, A322103.

Table[Function[k, Sum[d DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, d^(k+1)*sigma(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} j*sigma_k(j)*x^j/(1 - x^j).
L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^sigma_k(j)).
A(n,k) = Sum_{d|n} d^(k+1)*sigma_1(n/d).

A322263 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{d|n} 1/d^k.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 7, 2, 1, 17, 28, 21, 6, 4, 1, 33, 82, 73, 26, 2, 2, 1, 65, 244, 273, 126, 25, 8, 4, 1, 129, 730, 1057, 626, 7, 50, 15, 3, 1, 257, 2188, 4161, 3126, 697, 344, 85, 13, 4, 1, 513, 6562, 16513, 15626, 671, 2402, 585, 91, 9, 2, 1, 1025, 19684, 65793, 78126, 23725, 16808, 4369, 757, 13, 12, 6

Offset: 1

Ilya Gutkovskiy, Dec 01 2018

			Square array begins:
  1,    1,      1,        1,        1,          1,  ...
  2,  3/2,    5/4,      9/8,    17/16,      33/32,  ...
  2,  4/3,   10/9,    28/27,    82/81,    244/243,  ...
  3,  7/4,  21/16,    73/64,  273/256,  1057/1024,  ...
  2,  6/5,  26/25,  126/125,  626/625,  3126/3125,  ...
  4,    2,  25/18,      7/6,  697/648,    671/648,  ...

Index entries for sequences related to sigma(n)

Columns k=0..24 give A000005, A017665, A017667, A017669, A017671, A017673, A017675, A017677, A017679, A017681, A017683, A017685, A017687, A017689, A017691, A017693, A017695, A017697, A017699, A017701, A017703, A017705, A017707, A017709, A017711.
Denominators are in A322264.
Cf. A109974, A279394.

Table[Function[k, Numerator[DivisorSigma[-k, n]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[DivisorSigma[k, n]/n^k]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[SeriesCoefficient[Sum[x^j/(j^k (1 - x^j)), {j, 1, n}], {x, 0, n}]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} x^j/(j^k*(1 - x^j)) (for rationals Sum_{d|n} 1/d^k).
Dirichlet g.f. of column k: zeta(s)*zeta(s+k) (for rationals Sum_{d|n} 1/d^k).
A(n,k) = numerator of sigma_k(n)/n^k.

A322264 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{d|n} 1/d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 4, 1, 1, 16, 27, 16, 5, 1, 1, 32, 81, 64, 25, 1, 1, 1, 64, 243, 256, 125, 18, 7, 1, 1, 128, 729, 1024, 625, 6, 49, 8, 1, 1, 256, 2187, 4096, 3125, 648, 343, 64, 9, 1, 1, 512, 6561, 16384, 15625, 648, 2401, 512, 81, 5, 1, 1, 1024, 19683, 65536, 78125, 23328, 16807, 4096, 729, 10, 11, 1

Offset: 1

Ilya Gutkovskiy, Dec 01 2018

			Square array begins:
  1,    1,      1,        1,        1,          1,  ...
  2,  3/2,    5/4,      9/8,    17/16,      33/32,  ...
  2,  4/3,   10/9,    28/27,    82/81,    244/243,  ...
  3,  7/4,  21/16,    73/64,  273/256,  1057/1024,  ...
  2,  6/5,  26/25,  126/125,  626/625,  3126/3125,  ...
  4,    2,  25/18,      7/6,  697/648,    671/648,  ...

Index entries for sequences related to sigma(n)

Columns k=0..24 give A000012, A017666, A017668, A017670, A017672, A017674, A017676, A017678, A017680, A017682, A017684, A017686, A017688, A017690, A017692, A017694, A017696, A017698, A017700, A017702, A017704, A017706, A017708, A017710, A017712.
Numerators are in A322263.
Cf. A109974, A279394.

Table[Function[k, Denominator[DivisorSigma[-k, n]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[DivisorSigma[k, n]/n^k]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[SeriesCoefficient[Sum[x^j/(j^k (1 - x^j)), {j, 1, n}], {x, 0, n}]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} x^j/(j^k*(1 - x^j)) (for rationals Sum_{d|n} 1/d^k).
Dirichlet g.f. of column k: zeta(s)*zeta(s+k) (for rationals Sum_{d|n} 1/d^k).
A(n,k) = denominator of sigma_k(n)/n^k.

A377331 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k,n).

Original entry on oeis.org

1, 7, 58, 707, 11186, 219202, 5097205, 137036819, 4179577045, 142539843882, 5374034016858, 221930535785918, 9962431381720780, 482997720973917947, 25151350530268841003, 1400042027334939211235, 82960609980815501293708, 5213812927633674297808237, 346394632975721545946690108

Offset: 1

Vaclav Kotesovec, Oct 25 2024

nonn

Cf. A109974, A205812.

Table[Sum[Binomial[n, k] * DivisorSigma[n, k], {k, 1, n}], {n, 1, 20}]

{a(n)=sum(k=1, n, binomial(n, k)*sigma(k, n))}

a(n) ~ n^n / (sqrt(1 + LambertW(exp(-1))) * exp(n) * LambertW(exp(-1))^n).

A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.

Original entry on oeis.org

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

Offset: 1

Matthew Conroy, Mar 04 2025

nonn

sigma_k(n) is the sum of the k-th powers of the divisors of n.

a(n) exists since one can prove that for k > n*(log 2 + 1/2 log(n-1)), sigma_k sets a record at n.

			For n = 1, k = 0 is enough so a(1) = 0.
For n = 2, k = 0 works since sigma_0(2) = 2 > 1 = sigma_0(1) so a(2) = 0.
For n = 3, sigma_0(3) = 2 = sigma_0(2), but sigma_1(3) = 1^1+3^1 = 4 > 3 = sigma_1(2) > 1 = sigma_1(1) so a(3) = 1.
For n = 4, sigma_0(4) = 1^0+2^0+4^0 = 3 > 2 = sigma_0(3) = sigma_0(2) > 1 = sigma_0(1) so a(4) = 0.
For n = 5, sigma_0(5) = 2 = sigma_0(2) and sigma_1(5) = 6 < sigma_1(4) = 7 but sigma_2(5) = 26 > sigma_2(4) > sigma_2(3) > sigma_2(2) > sigma_2(1) so a(5) = 2.

Cf. A000203, A001157, A002093, A002182, A098475, A109974, A193988.

check(n,k) =  my(m=0);for(i=1,n-1, my(s=sigma(i,k)); if(s>m,m=s)); if(sigma(n,k)>m,return(1),return(0));
a(n) = my(ii=0); while(!check(n, ii), ii++);  ii;

a(n) = 0 precisely when n is highly composite number A002182.
a(n) = 1 precisely when n is highly abundant A002093 and not highly composite.
a(n) = 2 precisely when n is in A193988 and is not highly composite and is not highly abundant.
a(n) <= m if n < A098475(m). Empirically, it appears that a(A098475(m)) = m+1. - Pontus von Brömssen, Mar 16 2025

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A123926 Greatest common divisor of sigma_k(n) for all k >= 1.

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A319278 Square array sigma_k(n) read down antidiagonals: sum of the k-th powers of the divisors of n.

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A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

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A322263 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{d|n} 1/d^k.

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A322264 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{d|n} 1/d^k.

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A377331 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k,n).

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A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.

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