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

Author

Robert G. Wilson v

Keywords

• nonn
• nice

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)

Examples

			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
		

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.

Crossrefs

Cf. A000203, A003502, A005276, A028982.

Programs

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 *)

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

Extensions

Computed by Fred W. Helenius (fredh(AT)ix.netcom.com)

Extended by T. D. Noe, Dec 29 2011