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A003503
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The larger of a betrothed pair.
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9
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75, 195, 1925, 1648, 2295, 6128, 16587, 20735, 75495, 206504, 219975, 309135, 507759, 549219, 544784, 817479, 1057595, 1902215, 1331967, 1159095, 1763019, 1341495, 1348935, 1524831, 1459143, 2576945, 2226014, 2681019, 2142945, 2421704, 3220119, 3123735
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OFFSET
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1,1
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COMMENTS
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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
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)
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, B5.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Computed by Fred W. Helenius (fredh(AT)ix.netcom.com)
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STATUS
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approved
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