A003503 The larger of a betrothed pair.

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

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

Cf. A000203, A003502, A005276, A028982.

Sequence in context: A228306 A044407 A044788 * A201916 A098230 A348489

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