A015630 -id:A015630 - cp's OEIS frontend

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

Robert G. Wilson v

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)

			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

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

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.

Cf. A000203, A003502, A005276, A028982.

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 *)

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

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

Extended by T. D. Noe, Dec 29 2011

A007992 Augmented amicable pairs (smaller member of each pair).

Original entry on oeis.org

6160, 12220, 23500, 68908, 249424, 425500, 434784, 649990, 660825, 1017856, 1077336, 1238380, 1252216, 1568260, 1754536, 2166136, 2362360, 2482536, 2537220, 2876445, 3957525, 4177524, 4287825, 5224660, 5559510, 5641552

Offset: 1

N. J. A. Sloane

Let f(n) = 1 + sum of aliquot divisors of n; these are pairs (n,m) with f(n)=m, f(m)=n.
m cannot equal n. - Harvey P. Dale, May 18 2012
The term "augmented amicable numbers" was coined by Beck and Wajar (1977), who found the first 11 pairs. They also found the next 25 pairs (1993). - Amiram Eldar, Mar 09 2024

Amiram Eldar, Table of n, a(n) for n = 1..1159
Walter E. Beck and Rudolph M. Wajar, More reduced amicable pairs, Fibonacci Quarterly, Vol. 15, No. 4 (1977), pp. 331-332.
Walter E. Beck and Rudolph M. Wajar, Reduced and Augmented Amicable Pairs to 10^8, Fibonacci Quarterly, Vol. 31, No. 4 (1993), pp. 295-298.
David Moews, Augmented amicable pairs.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Paul Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq. 14 (2011), Article 11.5.2.
Eric Weisstein's World of Mathematics, Augmented Amicable Pair.

Cf. A015630.

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[ 6000000],aapQ]]] [[1]] (* Harvey P. Dale, May 18 2012 *)

A095702 Smallest "n-augmented" amicable number: the smallest positive integer k such that m = sigma(k) - k + n > k and k = sigma(m) - m + n, where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

220, 6160, 24, 180, 20, 6, 224, 2632, 40, 10, 16, 28, 340, 14, 15, 42, 66, 3696, 208, 20, 21, 54, 264, 24, 68, 26, 88, 120, 102, 30, 4030, 56, 33, 34, 35, 60, 110, 38, 280, 40, 354, 66, 476, 44, 130, 46, 408, 92, 1276, 96, 51, 52, 354, 78, 55, 120, 57, 58, 852, 60, 170

Offset: 0

Jack Brennen, Jul 06 2004

nonn

			a(1)= 6160 because sigma(6160)-6160+1 == 11697, sigma(11697)-11697+1 == 6160 and 6160 is the smallest integer for which this holds.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000203, A002025, A002046, A007992, A015630.

a[n_] := Module[{k = n + 1, s}, While[( s = DivisorSigma[1, k] - k + n) <= k || DivisorSigma[1, s] - s + n != k, k++]; k]; Array[a, 61, 0] (* Amiram Eldar, Dec 24 2020 *)

for(g=0,60,x=g+1;while(1,a=sigma(x)-x+g;if((a-x)*a,if(sigma(a)-a+g==x,print1(x,",");break));x+=1))

A306868 Larger of augmented bi-unitary amicable pair.

Original entry on oeis.org

871585, 1388145, 1483785, 2479065, 2580105, 4895241, 3830625, 7336455, 9100905, 10350345, 16933105, 9843526, 16367481, 24829945, 15706090, 18653745, 27866241, 21080865, 15439545, 23872185, 24401601, 32263905, 53763535, 63075321, 41337555, 60923577, 90245793

Offset: 1

Amiram Eldar, Mar 14 2019

nonn

A pair m < n is an augmented bi-unitary amicable pair if bsigma(m) = bsigma(n) = m + n - 1, where bsigma(n) is the sum of bi-unitary divisors of n (A188999).

The terms are ordered according to their lesser counterparts (A306867).

			871585 is in the sequence since it is the larger of the amicable pair (434784, 871585): bsigma(434784) = bsigma(871585) = 1306368 = 434784 + 871585 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..509 (terms with lesser member below 2*10^11, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers.

Cf. A015630, A126175, A188999, A292981, A306867.

fun[p_, e_]:=If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); f[n_] := bsigma[n] - n + 1; s={}; Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, m]], {n, 1, 10^7}]; s

A306873 Larger of augmented unitary amicable pair.

Original entry on oeis.org

7336455, 41337555, 110691295, 108212055, 154646206, 313439511, 6400149855, 9971007915, 10049576691, 9849706755, 12125842995, 12180547995, 14064001666, 18225635506, 26623431835, 20500208806, 23746912995, 23952459706, 43137954706, 56039259255, 99517314526, 125782774755

Offset: 1

Amiram Eldar, Mar 14 2019

nonn

A pair m < n is an augmented unitary amicable pair if usigma(m) = usigma(n) = m + n - 1, where usigma(n) is the sum of unitary divisors of n (A034460).

The terms are ordered according to their lesser counterparts (A306872).

			7336455 is in the sequence since it is the larger of the amicable pair (6224890, 7336455): usigma(6224890) = usigma(7336455) = 13561344 = 6224890 + 7336455 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..27 (terms with lesser member below 10^12, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers
J. O. M. Pedersen, Tables of Aliquot Cycles

Cf. A002953, A015630, A034460, A126175, A306872.

us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  s={}; Do[m = us[n] + 1; If[m > n && us[m] == n - 1, AppendTo[s, m]], {n, 1, 10^9}]; s

A126161 Number of augmented amicable pairs (m,n) with m

Original entry on oeis.org

0, 0, 0, 1, 4, 9, 36, 84, 188, 420, 930, 1931

Offset: 1

Ant King, Dec 20 2006

			a(6)=9 because there are 9 augmented amicable pairs with m<=10^6

Shyam Sunder Gupta, Amicable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 5, 159-183.
Jan Munch Pedersen, Known amicable pairs.

Cf. A007992, A015630.

s[n_]:=DivisorSigma[1,n]-n; AugmentedAmicableNumberQ[n_]:=If[s[s[n]+1]==n-1 && !DivisorSigma[1,n]==2n-1,True,False]; AugmentedAmicablePairList[ k_]:=(bnlist=Select[Range[k], AugmentedAmicableNumberQ[ # ]&]; newprlist= Table[Sort[{bnlist[[n]],s[bnlist[[n]]]+1}],{n,1,Length[bnlist]}]; augamprlist=Union[newprlist,newprlist]); data=AugmentedAmicablePairList[10^7]; Table[Length[Select[data,First[ # ]<10^k &]],{k,1,7}]

An augmented amicable pair (m,n) is a pair of integers m, n with m

A281265 Variation on betrothed numbers.

Original entry on oeis.org

6160, 11697, 12220, 16005, 23500, 28917, 68908, 76245, 249424, 339825, 425500, 434784, 570405, 649990, 660825, 678376, 697851, 871585, 1017856, 1077336, 1238380, 1252216, 1340865, 1483785, 1568260, 1754536, 1823925, 1899261, 2067625, 2166136, 2362360, 2479065

Offset: 1

Paolo P. Lava, Apr 13 2017

Members of a pair (x,y) such that sigma(x) = sigma(y) = x + y - 1, where sigma = A000203.

The first time a pair ordered by its first element is not adjacent is x = 425500, y = 570405 which correspond to a(11) and a(13), respectively.

			sigma(6160) = sigma(11697) = 6160 + 11697 - 1 = 17856.

Cf. A000203, A005276, A007992, A015630.

with(numtheory): P:=proc(q) local a,b,n; for n from 1 to q do
a:=sigma(n)-n+1; b:=sigma(a)-a+1; if b=n and a<>b then print(n);
fi; od; end: P(10^9);

A007992 U A015630.

Showing 1-7 of 7 results.

cp's OEIS Frontend

A003503 The larger of a betrothed pair.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A007992 Augmented amicable pairs (smaller member of each pair).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A095702 Smallest "n-augmented" amicable number: the smallest positive integer k such that m = sigma(k) - k + n > k and k = sigma(m) - m + n, where sigma(k) is the sum of the divisors of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A306868 Larger of augmented bi-unitary amicable pair.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A306873 Larger of augmented unitary amicable pair.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A126161 Number of augmented amicable pairs (m,n) with m

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A281265 Variation on betrothed numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula