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

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

Offset: 1

Author

Jaroslav Krizek, Mar 14 2018

Keywords

• nonn

Comments

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

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

Examples

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

Crossrefs

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

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Formula

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.