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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 (list; graph; refs; listen; history; text; internal format)
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, Quasi-amicable numbers, Math. Comp. 31 (1977), 608-611.

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]-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 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

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)