OFFSET
1,1
COMMENTS
It has been shown that (1) all known betrothed pairs are of opposite parity and (2) if a and b are a betrothed pair, and if a < b are of the same parity, then a > 10^10. See the reference for the Hagis & Lord paper. Can it be shown that all betrothed pairs are of opposite parity? - Harvey P. Dale, Apr 07 2013
From David A. Corneth, Jan 26 2019: (Start)
Let (k, m) be a betrothed pair. Then sigma(k) = sigma(m). Proof:
k = sigma(m) - m - 1 (1)
m = sigma(k) - k - 1 (2)
Partially substituting (1) in (2) gives
m = sigma(k) - (sigma(m) - m - 1) - 1 = sigma(k) - sigma(m) + m + 1 - 1 which simplifies to sigma(k) = sigma(m). QED.
If k and m are odd then they are both square. If k and m are even then they are square or twice a square (not necessarily both in the same family).
Proof: sigma(k) is odd iff k is a square or twice a square (cf. A028982). Hence if k isn't of that form (and sigma(k) is even) then the parity of sigma(k) - k - 1 is odd for odd k and even for even k.
If k is an odd square then sigma(k) - k - 1 is odd.
If k is twice a square or an even square then sigma(k) - k - 1 is even. QED.
Using inspection and the results above, if k and m are a betrothed pair of the same parity, the minimal term is > 2*10^14. (End)
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, B5.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..4122 (terms 1..1000 from Donovan Johnson, 1001..1126 from Amiram Eldar)
P. Hagis and G. Lord, Quasi-amicable numbers, Math. Comp. 31 (1977), 608-611.
David Moews, Augmented amicable pairs
Jan Munch Pedersen, Tables of Aliquot Cycles.
Wikipedia, Betrothed numbers.
EXAMPLE
75 is a term because sigma(75) - 75 - 1 = 124 - 75 - 1 = 48 and 75 > 48 and sigma(48) - 48 - 1 = 124 - 48 - 1 = 75. - David A. Corneth, Jan 24 2019
MATHEMATICA
aapQ[n_] := Module[{c=DivisorSigma[1, n]-1-n}, c!=n&&DivisorSigma[ 1, c]-1-c == n]; Transpose[Union[Sort[{#, DivisorSigma[1, #]-1-#}]&/@Select[Range[2, 10000], aapQ]]][[2]] (* Amiram Eldar, Jan 24 2019 after Harvey P. Dale at A015630 *)
PROG
(PARI) is(n) = m = sigma(n) - n - 1; if(m < 1 || n <= m, return(0)); n == sigma(m) - m - 1 \\ David A. Corneth, Jan 24 2019
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Computed by Fred W. Helenius (fredh(AT)ix.netcom.com)
Extended by T. D. Noe, Dec 29 2011
STATUS
approved