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A159907
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Numbers n with half-integral abundancy index, sigma(n)/n = k+1/2 with integer k.
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6
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2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, 8583644160, 10200236032, 21857648640, 57575890944, 57629644800, 206166804480, 17116004505600, 1416963251404800, 15338300494970880
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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.
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
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LINKS
| Walter Nissen, Abundancy : Some Resources
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FORMULA
| A159907 = { n | 2*A000203(n) is in n*A005408 } = { n | A054024(n) = n/2 }
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EXAMPLE
| a(1) = 2 since sigma(2)/2 = (1+2)/2 = 3/2 is of the form k+1/2 with integer k=1.
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.
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.
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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
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CROSSREFS
| Cf. A000203, A088912, A141643 (k=2), A055153 (k=3), A141645 (k=4), A159271 (k=5).
Sequence in context: A111430 A059332 A000794 * A088912 A203465 A055462
Adjacent sequences: A159904 A159905 A159906 * A159908 A159909 A159910
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KEYWORD
| nonn
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AUTHOR
| M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 25 2009
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