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A020487 Antiharmonic numbers: numbers n such that sigma_1(n) divides sigma_2(n). 21
1, 4, 9, 16, 20, 25, 36, 49, 50, 64, 81, 100, 117, 121, 144, 169, 180, 196, 200, 225, 242, 256, 289, 324, 325, 361, 400, 441, 450, 468, 484, 500, 529, 576, 578, 605, 625, 650, 676, 729, 784, 800, 841, 900, 961, 968, 980, 1024, 1025, 1058, 1089, 1156, 1225, 1280, 1296 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Numbers k such that antiharmonic mean of divisors of k is an integer. Antiharmonic mean of divisors of number m = Product (p_i^e_i) is A001157(m)/A000203(m) = Product ((p_i^(e_i+1)+1)/(p_i+1)). So a(n) = k, for some n, if  A001157(k)/A000203(k) is an integer. - Jaroslav Krizek, Mar 09 2009

Squares are antiharmonic, since (p^(2*e+1)+1)/(p+1) = p^(2*e) - p^(2*e-1) + p^(2*e-2) - ... + 1 is an integer. The nonsquare antiharmonic numbers are A227771. They include the primitive antiharmonic numbers A228023, except for its first term.  - Jonathan Sondow, Aug 02 2013

Sequence is infinite, see A227771. - Charles R Greathouse IV, Sep 02 2013

The term "antiharmonic" is also known as "contraharmonic". - Pahikkala Jussi, Dec 11 2013

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.

EXAMPLE

a(3) = 9 = 3^2; antiharmonic mean of divisors of 9 is (3^(2+1) + 1)/(3 + 1) = 7; 7 is integer. - Jaroslav Krizek, Mar 09 2009

MAPLE

with(numtheory);

List020487:=proc(q) local a, b, k, n;

for n from 1 to q do

  a:=divisors(n); b:=add(a[k]^2, k=1..nops(a));

  if type(b/sigma(n), integer) then print(n); fi;

od; end: List020487(10^6); # Paolo P. Lava, Apr 10 2013

MATHEMATICA

Select[Range[2000], Divisible[DivisorSigma[2, #], DivisorSigma[1, #]]&] (* Jean-Fran├žois Alcover, Nov 14 2017 *)

PROG

(MAGMA) [n: n in [1..1300] | IsZero(DivisorSigma(2, n) mod DivisorSigma(1, n))]; // Bruno Berselli, Apr 10 2013

(PARI) is(n)=sigma(n, 2)%sigma(n)==0 \\ Charles R Greathouse IV, Jul 02 2013

(Haskell)

a020487 n = a020487_list !! (n-1)

a020487_list = filter (\x -> a001157 x `mod` a000203 x == 0) [1..]

-- Reinhard Zumkeller, Jan 21 2014

CROSSREFS

Cf. A001157, A000203, A227771, A228023, A228024, A228036.

Sequence in context: A066213 A010441 A046871 * A046870 A313335 A313336

Adjacent sequences:  A020484 A020485 A020486 * A020488 A020489 A020490

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified March 21 01:18 EDT 2019. Contains 321356 sequences. (Running on oeis4.)