

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
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 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

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, Quasiamicable numbers, Math. Comp. 31 (1977), 608611.
D. 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]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

Cf. A000203, A003502, A005276, A028982.
Sequence in context: A228306 A044407 A044788 * A201916 A098230 A258056
Adjacent sequences: A003500 A003501 A003502 * A003504 A003505 A003506


KEYWORD

nonn,nice


AUTHOR

Robert G. Wilson v


EXTENSIONS

Computed by Fred W. Helenius (fredh(AT)ix.netcom.com)
Extended by T. D. Noe, Dec 29 2011


STATUS

approved



