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A159907 Numbers n with half-integral abundancy index, sigma(n)/n = k+1/2 with integer k. 18
2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, 8583644160, 10200236032, 21857648640, 57575890944, 57629644800, 206166804480, 17116004505600, 1416963251404800, 15338300494970880 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Obviously, all a(k) must be even (cf. formula), but e.g. a(9) and a(12) are not divisible by 3. See A007691 for numbers with integral abundancy.

Odd numbers and higher powers of 2 cannot be in the sequence; 6 is in A000396 and thus in A007691, and n=10,12,14,18,20,22 don't have integral 2*sigma(n)/n.

Conjecture: with number 1, multiply-anti-perfect numbers m: m divides antisigma(m) = A024816(m). Sequence of fractions antisigma(m) / m: {0, 0, 10, 2157, 2337, 13101, 4455356, ...}. - Jaroslav Krizek, Jul 21 2011

The above conjecture is equivalent to the conjecture that there are no odd multiply perfect numbers (A007691) greater than 1. Proof: (sigma(n)+antisigma(n))/n = (n+1)/2 for all n. If n is even then sigma(n)/n is a half-integer if and only if antisigma(n)/n is an integer. Since all members of this sequence are known to be even, the only way the conjecture can fail is if antisigma(n)/n is an integer, in which case sigma(n)/n is an integer as well. - Nathaniel Johnston, Jul 23 2011

LINKS

Table of n, a(n) for n=1..19.

Walter Nissen, Abundancy : Some Resources .

Project Euler, Problem 241: Perfection Quotients.

FORMULA

A159907 = { n | 2*A000203(n) is in n*A005408 } = { n | A054024(n) = n/2 }

EXAMPLE

a(1) = 2 since sigma(2)/2 = (1+2)/2 = 3/2 is of the form k+1/2 with integer k=1.

a(2) = 24 is in the sequence since sigma(24)/24 = (1+2+3+4+6+8+12+24)/24 = (24+12+24)/24 = k+1/2 with integer k=2.

MAPLE

with(numtheory);

P:=proc(i)

local a, n;

for n from 2 to i do

  a:=(n+1)/2-sigma(n)/n; if a=trunc(a) then print(n); fi;

od;

end:

P(10000000000); # Paolo P. Lava, Dec 12 2011

PROG

(PARI) isok(n) = denominator(sigma(n, -1)) == 2; \\ Michel Marcus, Sep 19 2015

CROSSREFS

Cf. A000203, A088912, A141643 (k=2), A055153 (k=3), A141645 (k=4), A159271 (k=5).

Sequence in context: A059332 A000794 * A242484 A088912 A203465 A055462

Adjacent sequences:  A159904 A159905 A159906 * A159908 A159909 A159910

KEYWORD

nonn

AUTHOR

M. F. Hasler, Apr 25 2009

STATUS

approved

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Last modified April 29 21:35 EDT 2017. Contains 285614 sequences.