A036471 -id:A036471 - cp's OEIS frontend

A386727 Numbers x such that there exist three integers 0

3, 10, 24, 51, 78, 105, 114, 136, 186, 220, 224, 255, 322, 348, 357, 370, 435, 478, 506, 616, 642, 710, 748, 820, 861, 885, 957, 996, 1004, 1068, 1113, 1214, 1221, 1276, 1292, 1336, 1390, 1485, 1491, 1562, 1564, 1581, 1605, 1660, 1670, 1704, 1716, 1724, 1815, 1869, 1880, 1912, 1947

Offset: 1

S. I. Dimitrov, Jul 31 2025

nonn

The numbers x, y, z and t form an amicable quadruple according to Yanney’s definition.

			114 is in the sequence since sigma(114) = sigma(158) = sigma(209) = sigma(239) = 240 = (114 + 158 + 209 + 239)/3.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Benjamin Franklin Yanney, Another definition of amicable numbers and some of their relations to Dickson's amicables, Amer. Math. Monthly, Vol. 30, No. 6 (1923), 311-315.

Cf. A000203, A036471, A386726.

isok(x1) = my(s=sigma(x1), vx=select(x->(x>=x1), invsigma(s)), v=vector(4, i, vx[1])); for (i=1, #vx, v[2] = vx[i]; for (j=1, #vx, v[3] = vx[j]; for (k=1, #vx, v[4] = vx[k]; if (vecsum(v) == 3*s, return(1));););); \\ Michel Marcus, Aug 01 2025

More terms from Michel Marcus, Aug 01 2025

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).

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A386727 Numbers x such that there exist three integers 0

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