

A003503


The larger of a betrothed pair.


9



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

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
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 square or twice a square (cf. A028982). Hence if 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 same parity, the minimal term is > 2*10^14. (End)


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B5.


LINKS



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]1n}, c!=n&&DivisorSigma[ 1, c]1c == 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)


STATUS

approved



