A161005 -id:A161005 - cp's OEIS frontend

A036474 Amicable quadruples: the numbers d referred to in A036471.

3834000, 4240800, 5089392, 5666760, 6939360, 7142100, 7932240, 8076960, 7976160, 8122464, 7738080, 7738920, 8136240, 8136240, 9121680, 9312480, 9368520, 9143820, 10159800, 10260180, 10403820, 10455984, 11179980, 10473120, 10930320, 10794960

Offset: 1

Yasutoshi Kohmoto

nonn

Donovan Johnson, Table of n, a(n) for n = 1..2000

Cf. A036471, A036472, A036473, A116148.
Cf. A002025, A002046 and A161005 for amicable pairs.
Cf. A125490 - A125492 and A137231 for amicable triples.
Cf. A233553 for amicable quintuples.

The present first term was found by Dean Hickerson, Nov 06 2006.
That this is the first term was confirmed by Giovanni Resta, Nov 14 2006, who also found a(2)-a(18).
Edited by N. J. A. Sloane, Nov 07 2006 and Nov 27 2006

A180164 The sum of the two numbers in an amicable pair, A002025(n) + A002046(n).

Original entry on oeis.org

504, 2394, 5544, 10584, 12600, 21600, 26880, 35712, 139104, 133920, 138240, 157248, 168480, 224640, 262080, 245520, 294840, 311040, 348192, 357120, 388800, 399168, 645624, 698544, 749952, 756000, 892800, 955206, 1017792, 1048320

Offset: 1

T. D. Noe, Aug 14 2010

nonn

This sequence initially shares many terms with A161005 because small amicable pairs are sometimes consecutive terms in the sorted list of amicable numbers, A063990.
This sequence is sorted by the smaller (abundant) member from A002025, so a(n) is not increasing. - Jeppe Stig Nielsen, Jan 27 2015
Duplicates occur, e.g., a(32)=a(35)=1296000. - Jeppe Stig Nielsen, Jan 27 2015
Comment originally by M. F. Hasler, Dec 14 2013, in A161005: "Also: The common value of sigma(a) = sigma(b) of the amicable pairs (a,b). See A137231 for the analog for amicable triples, and A116148 for quadruples." - Jeppe Stig Nielsen, Jan 27 2015
It is not known if a(n) is always even (see Hagis links). - Jeppe Stig Nielsen, Jan 31 2015
Are all terms abundant (A005101)? The first 10000 terms are. - Ivan N. Ianakiev, Apr 15 2021

			a(9) = A002025(9) + A002046(9) = 63020 + 76084 = 139104.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Peter Hagis, Lower bounds for relatively prime amicable numbers of opposite parity, Math. Comp. 24 (1970), 963-968.
Peter Hagis, Relatively Prime Amicable Numbers of Opposite Parity, Mathematics Magazine, Vol. 43, No. 1 (Jan., 1970), pp. 14-20.
Eric W. Weisstein's World of Mathematics, Pair Sum.

Cf. A002025, A002046, A066539, A259180 (amicable pairs).

s[n_] := DivisorSigma[1,n]-n; smallAmicableQ[n_] := Module[{b=s[n]}, n

Formula

a(n) = A259180(2n-1) + A259180(2n). - Omar E. Pol, Oct 22 2017

A259953 The sum (in nondecreasing order) of the two numbers in an amicable pair.

Original entry on oeis.org

504, 2394, 5544, 10584, 12600, 21600, 26880, 35712, 133920, 138240, 139104, 157248, 168480, 224640, 245520, 262080, 294840, 311040, 348192, 357120, 388800, 399168, 645624, 698544, 749952, 756000, 892800, 955206, 1017792, 1048320, 1270080, 1296000, 1296000, 1315440, 1347840, 1451520, 1522800, 1666560, 1781136, 1879200, 2041200

Offset: 1

Omar E. Pol, Jul 10 2015

nonn

Also the common value of sigma(x) = sigma(y) of the amicable pairs (x < y) ordered by nondecreasing sum (x + y). See A259933.
Duplicates occur, e.g., a(32) = a(33) = 1296000.
Another version of A180164.
First differs from both A161005 and A180164 at a(9).

			------------------------------------------
      A m i c a b l e   p a i r      Sum
------------------------------------------
n     A260086(n)  +  A260087(n)  =   a(n)
------------------------------------------
1         220            284          504
2        1184           1210         2394
3        2620           2924         5544
4        5020           5564        10584
5        6232           6368        12600
6       10744          10856        21600
7       12285          14595        26880
8       17296          18416        35712
9       66928          66992       133920
10      67095          71145       138240
11      63020          76084       139104
12      69615          87633       157248
...       ...            ...          ...
32     609928         686072      1296000
33     643336         652664      1296000
...

Cf. A000203, A063990, A161005, A180164, A259180, A259933, A260086, A260087.

a(n) = A259933(2n-1) + A259933(2n) = A260086(n) + A260087(n).

A180163 Products of pairs of amicable numbers (see A063990).

Original entry on oeis.org

62480, 1432640, 7660880, 27931280, 39685376, 116636864, 179299575, 318523136, 4217802560, 4494828240, 4952759175, 6067699000, 7775676090, 12285798525, 15069863936, 17358731325, 20160203840, 25845386480, 30293400832

Offset: 1

Jonathan Vos Post, Aug 14 2010

For a more reasonable sequence, in which both factors always belong to the same amicable pair, see A180202, which first differs from this sequence at a(9). - Omar E. Pol, Oct 25 2017

			a(1) = 220 * 284 = 62480 = 2^4 * 5 * 11 * 71.
a(2) = 1184 * 1210 = 1432640 = 2^6 * 5 * 11^2 * 37.

Cf. A002025, A002046, A063990, A161005, A180202, A259180.

a(n) = A063990(2*n-1) * A063990(2*n).

A179612 Sums of pairs of betrothed (or quasi-amicable) numbers.

Original entry on oeis.org

123, 335, 2975, 3223, 4319, 11903, 25479, 30239, 138239, 393119, 416639, 508895, 773759, 861839, 1071359, 1391039, 1645055, 2903039, 2413151, 2298239, 2903039, 2515199, 2557439, 2757887, 2695679, 3856895, 4147199, 4717439, 4245695, 4561919, 5391359, 5322239

Offset: 1

Jonathan Vos Post, Jan 08 2011

nonn

This is to A161005 sums of pairs of amicable numbers as betrothed (or quasi-amicable) numbers A005276 are to A063990 amicable numbers. The subsequence of primes begins: 11903, 138239, 1071359.

			a(1) = 48 + 75 = 123 = 3 * 41.
a(2) = 140 + 195 = 335 = 5 * 67.
a(6) = 5775 + 6128 = 11903 is the smallest prime in these pair-sums.

Cf. A000203, A003502, A003503, A005276, A063990, A161005.

a(n) = A003502(n) + A003503(n). {(j + k) such that sigma(j)=sigma(k)=j+k+1, where sigma=A000203}.

More terms from Amiram Eldar, Jan 27 2019

A180277 a(n) = A002952(n) + A002953(n).

Original entry on oeis.org

240, 2400, 40320, 72576, 94080, 120960, 141120, 311040, 403200, 483840, 483840, 725760, 673920, 1010880, 1209600, 1497600, 1411200, 1612800, 1747200, 2246400, 2177280, 2371200, 2449440, 2419200, 2620800, 2903040, 3144960, 3110400

Offset: 1

Jonathan Vos Post, Aug 23 2010

			a(1) = 114 + 126 = 240 = 2^4 * 3 * 5.
a(2) = 1140 + 1260 = 2400 = 2^5 * 3 * 5^2.
a(10) = a(11) - why?

Amiram Eldar, Table of n, a(n) for n = 1..7896

Cf. A002952, A002953, A161005, A180163.

a(n) = A002952(n) + A002953(n).

{(j+k) such that sigma*(j) = sigma*(k) = j+k, where sigma*(n) is the unitary divisor function A034448}.

Extended by R. J. Mathar, Aug 26 2010

Shorter name using given formula from Joerg Arndt, Jul 29 2024

A233626 Least member of an amicable n-tuple: (x[1],...,x[n]) such that sigma(x[1])=...=sigma(x[n])=x[1]+...+x[n], x[i]

Original entry on oeis.org

1, 220, 1980, 3270960, 53542288800

Offset: 1

M. F. Hasler, Dec 12 2013

Some authors use other definitions for amicable k-tuples, cf. link to MathWorld.

Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A002025, A002046 and A161005 for amicable pairs.
Cf. A125490 - A125492 and A137231 for amicable triples.
Cf. A036471 - A036474 and A116148 for amicable quadruples.
Cf. A233553 for amicable quintuples.

A233538 Triangle T(n,k) read by rows, which contains for 1<=k<=n the least amicable n-tuple T(n,1),..., T(n,n) such that sigma(T(n,k)) = T(n,1)+...+T(n,n).

Original entry on oeis.org

1, 220, 284, 1980, 2016, 2556, 3270960, 3361680, 3461040, 3834000, 53542288800, 59509850400, 59999219280, 60074174160, 61695597600

Offset: 1

Michel Marcus, M. F. Hasler, Dec 11 2013

Like amicable pairs, amicable n-tuples can be regular or irregular (see Pedersen link). The first amicable pair is regular. Then the first n-tuples are irregular.
For n=3 to 5, the first regular n-tuples are: [230880, 267168, 306336], [6966960, 7054320, 7840560, 8136240], [55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440].
On the other hand, for n>2, a n-tuple can be "very" irregular, that is, when the values of sigma(n-tuple[i]/GCD(n-tuple)) are all different. The first such n-tuples are [21168, 22200, 27312], [3767400, 4090320, 4150440, 4240800].
When n=2, irregular and "very irregular" is the same thing. The first irregular amicable pair is (1184, 1210) (see difference between A002025 and A215491).
Regular n-tuples can be found with the method described in the second Kohmoto link. Then it is eventually possible to derive another n-tuple using the same "seed". For this, it suffices to find an integer g' such that sigma(g')/g' = sigma(g)/g and coprime to the terms of the n-tuple divided by g.
The 6th row is smaller than (379952828833009557565440000, 387198605857900590673920000, 388674597474082097418240000, 388808778530098598031360000, 389307165309588457451520000, 393332596990083475845120000).

			Triangle begins:
1;
220, 284;                                 i.e. A002025(1), A002046(1).
1980, 2016, 2556;                         i.e. A125490(1), A125491(1), A125492(1).
3270960, 3361680, 3461040, 3834000;
53542288800, 59509850400, 59999219280, 60074174160, 61695597600.

Donovan Johnson, Re: A_k and RMPN, SeqFan list, Dec 12 2013
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, A_k and RMPN, SeqFan list, Dec 09 2013
J. M. Pedersen, Type system of amicable pairs
Eric Weisstein's World of Mathematics, Amicable Pair
Eric Weisstein's World of Mathematics, Amicable Triple
Eric Weisstein's World of Mathematics, Amicable Quadruple

Cf. A233626 (first column).
Cf. A002025, A002046, A161005, (amicable pairs).
Cf. A125490 - A125492, A137231, (amicable triples).
Cf. A036471 - A036474, A116148, (amicable quadruples).
Cf. A233553, A233626 (first row).

cp's OEIS Frontend

A036474 Amicable quadruples: the numbers d referred to in A036471.

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A180164 The sum of the two numbers in an amicable pair, A002025(n) + A002046(n).

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A259953 The sum (in nondecreasing order) of the two numbers in an amicable pair.

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A180163 Products of pairs of amicable numbers (see A063990).

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A179612 Sums of pairs of betrothed (or quasi-amicable) numbers.

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A180277 a(n) = A002952(n) + A002953(n).

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A233626 Least member of an amicable n-tuple: (x[1],...,x[n]) such that sigma(x[1])=...=sigma(x[n])=x[1]+...+x[n], x[i]

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A233538 Triangle T(n,k) read by rows, which contains for 1<=k<=n the least amicable n-tuple T(n,1),..., T(n,n) such that sigma(T(n,k)) = T(n,1)+...+T(n,n).

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