A100879 -id:A100879 - cp's OEIS frontend

A300906 Numbers k such that sigma(k)^k divides k^sigma(k).

1, 6, 28, 84, 120, 364, 420, 496, 672, 840, 1080, 1320, 1488, 1782, 2280, 2760, 3276, 3360, 3472, 3480, 3720, 3780, 5640, 7080, 7392, 7440, 7560, 8128, 8736, 9240, 9480, 10416, 10920, 11880, 12400, 15456, 15960, 16368, 16380, 17880, 18360, 18600, 19320, 20520

Offset: 1

Jaroslav Krizek, Mar 20 2018

nonn

Numbers k such that A217872(k) divides A100879(k).
Numbers k such that A300905(k) = 0.
Corresponding quotients: 1, 729, 123476695691247935826229781856256, ...
m-perfect numbers k (A007691) are terms iff m divides k.

			6 is a term because 6^sigma(6) / sigma(6)^6 = 6^12 / 12^6 = 2176782336 / 2985984 = 729 (integer).

Cf. A000203, A100879, A217872, A300905.

Filtered([1..30000],n->PowerModInt(n,Sigma(n),Sigma(n)^n)=0); # Muniru A Asiru, Mar 20 2018

[n: n in[1..20000]  | n^SumOfDivisors(n) mod SumOfDivisors(n)^n eq 0];

with(numtheory):
select(n->n &^ sigma(n) mod sigma(n)^n =0, [`$`(1..30000)]); # Muniru A Asiru, Mar 20 2018

isok(n) = my(s = sigma(n)); Mod(n, s^n)^s == 0; \\ Michel Marcus, Mar 23 2018

A165797 a(n) = n^( sigma(n) - tau(n) ).

Original entry on oeis.org

1, 2, 9, 256, 625, 1679616, 117649, 8589934592, 3486784401, 100000000000000, 25937424601, 552061438912436417593344, 23298085122481, 83668255425284801560576, 332525673007965087890625

Offset: 1

Jaroslav Krizek, Sep 27 2009

nonn

The power of n with exponent given by the difference between its sum of divisors and its count of divisors.

			a(4) = 4^(sigma(4)-tau(4)) = 4^(7-3) = 4^4 = 256.

Jaroslav Krizek, Table of n, a(n) for n = 1..50

Table[n^[ DivisorSigma[1, n] - DivisorSigma[0, n]], {n, 50}]

a(n) = n^(A000203(n)-A000005(n)) = n^A000203(n) / n^A000005(n) = n^A065608(n).
a(n) = A100879(n) / A062758(n).
a(p) = p^(p-1) for p = prime.

Slightly edited by R. J. Mathar, Sep 29 2009

A300905 a(n) = n^sigma(n) mod sigma(n)^n.

Original entry on oeis.org

0, 8, 17, 1978, 73, 0, 1570497, 1009588832, 7390478182, 1391503283200, 166394893969, 151448237549551616, 762517292682713, 18685202394240778240, 814227337406354049, 187036938412352867328077, 947615093635545799201, 2095989269871299377743863001

Offset: 1

Jaroslav Krizek, Mar 14 2018

nonn

sigma(n) = the sum of the divisors of n (A000203).

n^sigma(n) > sigma(n)^n for all n > 2.

			For n = 6; a(6) = 0 because 6^sigma(6) mod sigma(6)^6 = 6^12 mod 12^6 = 2176782336 mod 2985984 = 0.

Cf. A000203, A007691, A100879, A217872, A300906.

List([1..20],n->PowerModInt(n,Sigma(n),Sigma(n)^n))); # Muniru A Asiru, Mar 20 2018

[n^SumOfDivisors(n) mod SumOfDivisors(n)^n: n in[1..20]];

with(numtheory): seq(n &^ sigma(n) mod sigma(n)^n,n=1..20); # Muniru A Asiru, Mar 20 2018

Array[With[{s = DivisorSigma[1, #]}, PowerMod[#, s, s^#]] &, 18] (* Michael De Vlieger, Mar 16 2018 *)

a(n) = my(s=sigma(n)); lift(Mod(n, s^n)^s); \\ Michel Marcus, Mar 17 2018

a(n) = A100879(n) mod A217872(n).
a(n) = 0 for numbers n in A300906.
If n is a k-perfect number from A007691, then a(n) = 0 iff k divides n.

A176030 a(n) = n^(sigma_2(n)).

Original entry on oeis.org

1, 32, 59049, 4398046511104, 1490116119384765625, 808281277464764060643139600456536293376, 1798465042647412146620280340569649349251249

Offset: 1

Michel Lagneau, Apr 06 2010

sigma_2 = A001157 is the sum of squares of divisors of n.

			a(1) = 1^1 = 1.
a(2) = 2^(1+4) = 32.
a(3) = 3^(1+9) = 59049.

Cf. A001157, A100879.

Mathematica
```
Table[n^DivisorSigma[2,n], {n, 20}]
```

a(n) = n^A001157(n).

cp's OEIS Frontend

A300906 Numbers k such that sigma(k)^k divides k^sigma(k).

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A165797 a(n) = n^( sigma(n) - tau(n) ).

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A300905 a(n) = n^sigma(n) mod sigma(n)^n.

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A176030 a(n) = n^(sigma_2(n)).

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