A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

1, 6, 30, 66, 102, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, 672, 690, 714, 840, 870, 924, 930, 966, 1122, 1320, 1410, 1428, 1518, 1590, 1638, 1722, 1770, 1890, 1932, 2130, 2226, 2280, 2310, 2346, 2370, 2670, 2730, 2760, 2838, 2970, 2982, 3102, 3162, 3210, 3360, 3444, 3486, 3498, 3570, 3720, 3780, 3948, 3990

Offset: 1

Author

Ilya Gutkovskiy, Oct 19 2016

Keywords

• nonn

Comments

Intersection of A003601, A120736 and A145551.

Numbers k such that A000005(k)|A007955(k), A000203(k)|A007955(k) and A000005(k)| A000203(k).

Numbers k such that A000005(k)|A062981(k), A072861(k)|A062758(k) and A245656(k) = 1.

Examples

			a(2) = 6 because 6 has 4 divisors {1,2,3,6}, 1*2*3*6/4 = 9, 1*2*3*6/(1 + 2 + 3 + 6) = 3 and (1 + 2 + 3 + 6)/4 = 3 are integer.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..20000
• Eric Weisstein's World of Mathematics, Divisor Product.
• Eric Weisstein's World of Mathematics, Divisor Function.
• Wikipedia, Arithmetic number.

Crossrefs

Cf. A000005, A000203, A003601, A007955, A062758, A062981, A072861, A120736, A145551, A245656.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,k,n;for n from 1 to q do
a:=divisors(n); b:=mul(a[k],k=1..nops(a));
if type(sigma(n)/tau(n),integer) and type(b/sigma(n),integer) and
type(b/tau(n),integer) then print(n); fi;
od; end: P(10^5); # Paolo P. Lava, Oct 20 2016

Mathematica

Select[Range[4000], Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[1, #1]] && Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[0, #1]] && Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] & ]