A003503 - OEIS

%I #64 Sep 08 2023 22:52:59

%S 75,195,1925,1648,2295,6128,16587,20735,75495,206504,219975,309135,

%T 507759,549219,544784,817479,1057595,1902215,1331967,1159095,1763019,

%U 1341495,1348935,1524831,1459143,2576945,2226014,2681019,2142945,2421704,3220119,3123735

%N The larger of a betrothed pair.

%C 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

%C From _David A. Corneth_, Jan 26 2019: (Start)

%C Let (k, m) be a betrothed pair. Then sigma(k) = sigma(m). Proof:

%C k = sigma(m) - m - 1 (1)

%C m = sigma(k) - k - 1 (2)

%C Partially substituting (1) in (2) gives

%C m = sigma(k) - (sigma(m) - m - 1) - 1 = sigma(k) - sigma(m) + m + 1 - 1 which simplifies to sigma(k) = sigma(m). QED.

%C 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).

%C 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.

%C If k is an odd square then sigma(k) - k - 1 is odd.

%C If k is twice a square or an even square then sigma(k) - k - 1 is even. QED.

%C 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)

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

%H Giovanni Resta, <a href="/A003503/b003503.txt">Table of n, a(n) for n = 1..4122</a> (terms 1..1000 from Donovan Johnson, 1001..1126 from Amiram Eldar)

%H P. Hagis and G. Lord, <a href="https://doi.org/10.1090/S0025-5718-1977-0434939-3">Quasi-amicable numbers</a>, Math. Comp. 31 (1977), 608-611.

%H David Moews, <a href="http://djm.cc/augmented.fmtlist">Augmented amicable pairs</a>

%H Jan Munch Pedersen, <a href="http://62.198.248.44/aliquot/tables.htm">Tables of Aliquot Cycles</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Betrothed_numbers">Betrothed numbers</a>.

%e 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

%t 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 *)

%o (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

%Y Cf. A000203, A003502, A005276, A028982.

%K nonn,nice

%O 1,1

%A _Robert G. Wilson v_

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

%E Extended by _T. D. Noe_, Dec 29 2011