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A277521 Numbers n such that number of divisors of n and sum of divisors of n divides product of divisors of n and the average of the divisors of n is an integer. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Intersection of A003601, A120736 and A145551.

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

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

LINKS

Table of n, a(n) for n=1..58.

Eric Weisstein's World of Mathematics, Divisor Product

Eric Weisstein's World of Mathematics, Divisor Function

Wikipedia, Arithmetic number

EXAMPLE

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.

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]] & ]

CROSSREFS

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

Sequence in context: A145010 A056835 A056836 * A163640 A199130 A152743

Adjacent sequences:  A277518 A277519 A277520 * A277522 A277523 A277524

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Oct 19 2016

STATUS

approved

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Last modified February 24 01:16 EST 2020. Contains 332195 sequences. (Running on oeis4.)