A090615 -id:A090615 - cp's OEIS frontend

A180624 Partials sums of A090615.

1264460, 3379784, 6164364, 11102500, 18271604, 36320580, 54976960, 83135125, 129857825, 210986457, 385264277, 594788487, 924792067, 1423007483, 2659409715, 4458691045, 6846467595, 9563962830, 12443660134, 16149431959

Offset: 1

Jonathan Vos Post, Sep 13 2010

Partials sums of smallest member of sociable quadruples. Primes in this partial sum begin: 924792067, 16149431959.

			a(13) = 1264460 + 2115324 + 2784580 + 4938136 + 7169104 + 18048976 + 18656380 + 28158165 + 46722700 + 81128632 + 174277820 + 209524210 + 330003580 = 924792067 is prime.

Eric W. Weisstein, Sociable Numbers

Cf. A090615.

a(n) = sum[i=1..n] A090615(i).

A319902 Unitary sociable numbers of order 4.

Original entry on oeis.org

263820, 263940, 280380, 280500, 395730, 395910, 420570, 420750, 172459210, 209524210, 218628662, 218725430, 230143790, 231439570, 246667790, 272130250, 384121920, 384296640, 408233280, 408408000

Offset: 1

Michel Marcus, Oct 01 2018

Is this a duplicate of A098188? - R. J. Mathar, Oct 04 2018
Note that the first 4 terms and the next 4 terms form two sociable groups. But then the next 8 terms belong to two distinct sociable groups, whereas in A098188 the integers are grouped by cycle.
From Hartmut F. W. Hoft, Aug 23 2023: (Start)
This sequence is A098188 in ascending order.
Among the 19 4-cycles listed in the link by J. O. M. Pedersen only four of the 6 possible patterns of relative sizes of the numbers in a cycle are realized. (End)

J. O. M. Pedersen, Known Unitary Sociable Numbers of order four [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Order 4 cycles, 2007.
Eric Weisstein's World of Mathematics, Unitary Sociable Numbers

Cf. A063919 (sum of proper unitary divisors).
Cf. A002827 (unitary perfect), A063991 (unitary amicable).
Cf. A097024 (order 5), A097030 (order 14).
Cf. A090615 (least member of sociable quadruples).
Cf. A098188.

f[n_] := f[n] = Module[{s = 0}, s = Total[Select[Divisors[n], GCD[#, n/#] == 1 &]]; Return[s - n]]; isok1[n_] := isok1[n] = Quiet[Check[f[n] == n, 0]]; isok2[n_] := isok2[n] = Quiet[Check[f[f[n]] == n, 0]]; isok4[n_] := isok4[n] = Quiet[Check[f[f[f[f[n]]]] == n, 0]]; isok[n_] := isok[n] = isok4[n] && Not[isok1[n]] && Not[isok2[n]]; Monitor[Position[Table[isok[n], {n, 1, 408408000}], True], n] (* Robert P. P. McKone, Aug 24 2023 *)

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok4(n) = iferr(f(f(f(f(n)))) == n, E, 0);
isok2(n) = iferr(f(f(n)) == n, E, 0);
isok1(n) = iferr(f(n) == n, E, 0);
isok(n) = isok4(n) && !isok1(n) && !isok2(n);

A222977 Smallest member of sociable quadruples with signature: abundant, deficient, abundant, deficient.

Original entry on oeis.org

18048976, 46722700, 4424606020, 73636082872, 88585861815, 90568599176, 518833084192, 1138168194296, 5793575538465

Offset: 1

Michel Marcus, Mar 11 2013

See comments in A222975.

1216327163895122388615 and 12015582455500352695694 are also terms. - Amiram Eldar, Apr 05 2024

			The first quadruple of this type is: 18048976, 20100368, 18914992, 19252208, that are respectively abundant, deficient, abundant, deficient.

Jan Munch Pedersen, Known Sociable Numbers of order four, Tables of Aliquot Cycles.

Cf. A090615, A222975, A222976, A222978.

a(8)-a(9) from Amiram Eldar, Apr 05 2024

A072892 The 4-cycle of the n => sigma(n)-n process. sigma(n) is the sum of divisors of n. (A000203).

Original entry on oeis.org

1264460, 1547860, 1727636, 1305184, 1264460

Offset: 1

Miklos Kristof, Jul 29 2002

The two smallest members of sociable quadruples (1264460 and 2115324, see A090615) were found by the Canadian mathematician and educator Kenneth Dudley Fryer (1924-1984) in 1965 (Honsberger, 1970). These were the first aliquot cycles of length greater than 2 that were found since 1918 (see A072890 and A072891). They were rediscovered by Cohen (1970). - Amiram Eldar, Mar 24 2024

Ross Honsberger, Ingenuity in Mathematics, Mathematical Association of America, 1970.

Henri Cohen, On amicable and sociable numbers, Math. Comp., Vol. 24, No. 110 (1970), pp. 423-429.
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number.

Cf. A000203, A001065, A072890, A072891, A090615.

NestWhileList[DivisorSigma[1, #] - # &, 1264460, UnsameQ, All] (* Amiram Eldar, Mar 24 2024 *)

a(4+n) = a(n).

A222975 Smallest member of sociable quadruples with signature: abundant, abundant, deficient, deficient.

Original entry on oeis.org

1264460, 2115324, 2784580, 7169104, 18656380, 28158165, 81128632, 174277820, 498215416, 1236402232, 1799281330, 2387776550, 2717495235, 2879697304, 15837081520, 17616303220, 91411869465, 111375706442, 661126361272, 741158497112, 1045805730255, 1092162882824

Offset: 1

Michel Marcus, Mar 11 2013

nonn

Members of a sociable quadruple are usually listed in the order with which they appear in the corresponding aliquot sequence. Therefore, in a sociable quadruple the 1st member is abundant, and the 4th member is deficient. In all, we can get 4 possibilities, and a sociable quadruple can be (with A for abundant, and D for deficient) : AADD, ADDD, ADAD, AAAD. The numbers in these sequences have been voluntarily limited to 12 digits maximum, the current status of the exhaustive search (see Pedersen link).

			The first quadruple of this type is: 1264460, 1547860, 1727636, 1305184, that are respectively abundant, abundant, deficient, deficient.

Amiram Eldar, Table of n, a(n) for n = 1..28
Jan Munch Pedersen, Known Sociable Numbers of order four, Tables of Aliquot Cycles.

Cf. A090615, A222976, A222977, A222978.

a(21)-a(22) from Amiram Eldar, Apr 05 2024

A222976 Smallest member of sociable quadruples with signature: abundant, deficient, deficient, deficient.

Original entry on oeis.org

4938136, 209524210, 4823923384, 8653956136, 21669628904, 44379752648, 3681459083984, 4553100850815

Offset: 1

Michel Marcus, Mar 11 2013

See comments in A222975.

			The first quadruple of this type is: 4938136, 5753864, 5504056, 5423384, that are respectively abundant, deficient, deficient, deficient.

Jan Munch Pedersen, Known Sociable Numbers of order four, Tables of Aliquot Cycles.

Cf. A090615, A222975, A222977, A222978.

a(7)-a(8) from Amiram Eldar, Apr 05 2024

A222978 Smallest member of sociable quadruples with signature: abundant, abundant, abundant, deficient.

Original entry on oeis.org

330003580, 3705771825, 5373457070

Offset: 1

Michel Marcus, Mar 11 2013

See comments in A222975.

a(4) > 10^14, if it exists. - Amiram Eldar, Apr 05 2024

			The first quadruple of this type is: 330003580, 363003980, 399304420, 440004764 that are respectively abundant, abundant, abundant, deficient.

Jan Munch Pedersen, Known Sociable Numbers of order four, Tables of Aliquot Cycles.

Cf. A090615, A222975, A222976, A222977.

A292217 Conjectured list of numbers in increasing order that belong to sociable cycles of length greater than 2 in which the sum of the cycle is divisible by 10.

Original entry on oeis.org

1264460, 1305184, 1547860, 1727636, 4938136, 5423384, 5504056, 5753864, 18656380, 20522060, 24289964, 28158165, 28630036, 29902635, 29971755, 30853845, 81128632, 91314968, 91401368, 96389032, 209524210, 230143790, 231439570, 246667790, 498215416, 506040584, 510137384, 583014136

Offset: 1

Zoltan Galantai, Sep 11 2017

This list is not known to be complete (564 might be a member). See A122726. - N. J. A. Sloane, Sep 17 2021

Up to the known 1593 sociable number cycles, 96.1% of the sociable number cycles satisfy this condition (up to the first 10 sociable number cycles: 40%; up to the first 100 sociable number cycles: 77%; up to the first 500 sociable number cycles: 92%, and up to the first 1000 sociable number cycles: 94.9%). So the conjecture here is that as the number of sociable number cycles increases, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%. Notice that the sums of amicable pairs are similarly often divisible by 10, but are not included here (see A291422).

			The sum of 1264460, 1547860, 1727636 and 1305184 is divisible by ten, thus this sociable number cycle belongs to the sequence. On the other hand, the 12496, 14288, 15472, 14536, 14264 sociable number cycle does not qualify since its sum is 71506.

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 62 - 63.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 2747 - 2748.
Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computation Approach, World Scientific 1996, pp. 34 - 38.

Zoltan Galantai, Table of n, a(n) for n = 1..36
Zoltan Galantai, List of known sociable number cycles where the sums of the cycles is divisible by 10.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
David Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]
Eric Weisstein's World of Mathematics, Sociable Numbers

Cf. A122726, A003416, A090615, A119478, A098007, A126016, A063990, A291422.

Changed definition and added comment to point out that this sequence is only conjectural. - N. J. A. Sloane, Sep 17 2021

A319915 Smallest member of bi-unitary sociable quadruples.

Original entry on oeis.org

162, 1026, 1620, 10098, 10260, 41800, 51282, 100980, 107920, 512820, 1479006, 4612720, 4938136, 14790060, 14800240, 23168840, 28158165, 32440716, 55204500, 81128632, 84392560, 88886448, 209524210, 283604220, 325903500, 498215416, 572062304, 881697520

Offset: 1

Amiram Eldar, Oct 01 2018

nonn

The bi-unitary version of A090615.

			162 is in the sequence since the iterations of the sum of bi-unitary proper divisors function (A188999(n) - n) are cyclic with a period of 4: 162, 174, 186, 198, 162, ... and 162 is the smallest member of the quadruple.

Cf. A090615, A188999, A292980, A292981, A319902.

fun[p_, e_]:=If[Mod[e, 2]==1, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)];
bs[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]-n;seq[n_]:=NestList [bs, n,4][[2;;5]] ;aQ[n_] := Module[ {s=seq[n]}, n==Min[s] && Count[s,n]==1]; Do[If[aQ[n],Print[n]],{n,1,10^9}]

fn(n) = {if (n==1, 1, f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f) - n;);}
isok(n) = my(v = vector(5)); v[1] = n; for(k=2, 5, v[k] = fn(v[k-1])); (v[5] == n) && (vecmin(v) == n) && (#vecsort(v,,8)==4); \\ Michel Marcus, Oct 02 2018

is(n) = my(c = n); for(i = 1, 3, c = fn(c); if(c <= n, return(0))); c = fn(c); c == n \\ uses Michel Marcus' fn David A. Corneth, Oct 02 2018

A073032 A 4-cycle of the k => sigma(k)-k process, where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

18048976, 20100368, 18914992, 19252208, 18048976, 20100368, 18914992, 19252208, 18048976, 20100368, 18914992, 19252208, 18048976, 20100368, 18914992, 19252208, 18048976, 20100368, 18914992, 19252208, 18048976, 20100368, 18914992, 19252208, 18048976

Offset: 1

Miklos Kristof, Aug 22 2002

18048976 = 2^4*11*102551, 20100368 = 2^4*919*1367, 18914992 = 2^4*37*89*359, 19252208 = 2^4*1203263.

The cycle was discovered by Cohen (1970). - Amiram Eldar, Aug 18 2024

Henri Cohen, On amicable and sociable numbers, Math. Comp., Vol. 24, No. 110 (1970), pp. 423-429.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A000203, A045477, A073033, A090615.

a(n)=[19252208, 18048976, 20100368, 18914992][n%4+1] \\ Charles R Greathouse IV, Feb 19 2013

Terms corrected by Charles R Greathouse IV, Feb 19 2013

cp's OEIS Frontend

A180624 Partials sums of A090615.

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A319902 Unitary sociable numbers of order 4.

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A222977 Smallest member of sociable quadruples with signature: abundant, deficient, abundant, deficient.

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A072892 The 4-cycle of the n => sigma(n)-n process. sigma(n) is the sum of divisors of n. (A000203).

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A222975 Smallest member of sociable quadruples with signature: abundant, abundant, deficient, deficient.

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A222976 Smallest member of sociable quadruples with signature: abundant, deficient, deficient, deficient.

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A222978 Smallest member of sociable quadruples with signature: abundant, abundant, abundant, deficient.

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A292217 Conjectured list of numbers in increasing order that belong to sociable cycles of length greater than 2 in which the sum of the cycle is divisible by 10.

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A319915 Smallest member of bi-unitary sociable quadruples.

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