Zoltan Galantai - cp's OEIS frontend

A360694 Numbers whose divisors can be partitioned into two disjoint sets where the sum of both sets is prime.

4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 52, 54, 55, 56, 57, 58, 60, 63, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 99, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115

Offset: 1

Zoltan Galantai, Feb 17 2023

nonn

The concept of this sequence is similar to the concept of Zumkeller numbers (A083207) partitioning the sums of the divisors (A000203) into two sets.
This concept can be extended, since the sums of some numbers' divisors can be partitioned into more sets, e.g., 6 (2,3,7) and 10 (2,5,11) into three.
Some numbers can be divided more than one way. For 10, there are two divisons: (5,13) and (7,11) and for 20, there are four: (5,37), (11,31), (13,29) and (19,23).
From Robert Israel, Feb 21 2023: (Start)
Contains no primes.
k in A028982 is in the sequence iff k is even and A000203(k)-2 is prime.
(End)

			4 belongs to the sequence, since its divisors can be partitioned into two sets where the sums of these sets are primes (2,5). 9 does not belong to the sequence, because its divisors cannot be partitioned in this way.

Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc., 1996, p. 11, p. 22.

Yuejian Peng and K. P. S. Bhaskara Rao, On Zumkeller numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155
Eric Weisstein's World of Mathematics, Perfect Number.
Eric Weisstein's World of Mathematics, Prime.

Cf. A000040, A000396, A028982, A083207, A000203.

filter:= proc(n) local P,p,S,s;
  S:= numtheory:-divisors(n); s:= convert(S,`+`);
  P:= combinat:-subsets(S minus {n});
  while not P[finished] do
    p:= convert(P[nextvalue](),`+`);
    if isprime(p) and isprime(s-p) then return true fi
  od;
  false
end proc:
select(filter, [$1..200]); # Robert Israel, Feb 21 2023

q[n_] := Module[{d = Divisors[n], s, p}, s = Total[d]; p = Position[Rest @ CoefficientList[Product[1 + x^i, {i, d}], x], ?(# > 0 &)] // Flatten; AnyTrue[p, PrimeQ[#] && PrimeQ[s - #] &]]; Select[Range[100], q] (* _Amiram Eldar, Feb 18 2023 *)

A360140 Odd amicable pairs with only one member divisible by 3.

Original entry on oeis.org

445953248528881275, 659008669204392325, 748174019876835825, 906104451346869775, 1097581690986390225, 1615281291559017775, 1281431098689616875, 1769164614201263125, 1968382462511781225, 2982869282783783575, 1993991197249826775, 2901232579265245225, 2247817805416685775, 2726235257034514225

Offset: 1

Zoltan Galantai, Jan 26 2023

nonn

In these pairs only the smaller number is divisible by 3 and both the smaller and the larger number are divisible by 5, 7, and 25. These pairs prove that there are odd amicable pairs where only one of the numbers is divisible by 3.

			(445953248528881275, 659008669204392325) belongs to the sequence since the smaller number is divisible by 3 and the larger is not. The (12285, 14595) pair is not a part of the sequence, since both of its members are divisible by 3.

Richard K. Guy, Unsolved Problems in Number Theory. Springer, 2004, page 88.

S. Battiato and W. Borho, Are There Odd Amicable Numbers Not Divisible by Three, Mathematics of Computation, Apr., 1988, Vol. 50, No. 182, pp. 633-637.
S. Chernykh, Amicable pairs where the two members have different smallest prime factors, January 30th, 2016.
Zoltan Galantai, List of 31 known pairs

Cf. A259180, A002025, A002046, A262623.

A360054 Number of odd amicable pairs where the smaller term of the pair is less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 3, 8, 21, 55, 154, 412, 1088, 2632, 6532, 15371, 35218, 79982, 180061, 402560, 894404, 1975742

Offset: 1

Zoltan Galantai, Jan 23 2023

The list starts with n=1.

Comparing with the numbers of even amicable pairs in A066873, up to 10^4, the proportion of odd amicable pairs is 0%; up to 10^5 it is 23% and up to 10^10 is 28.9%. Up to 10^15, it is 40.4% and up to 10^19 this percentage is 45.9%. It is possible that this trend holds true for more amicable pairs, and thus most amicable number pairs are odd.

Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computational Approach, World Scientific, 1996, pages 151 - 153.

Sergei Chernykh, Amicable pairs list
Zoltan Galantai, Number of odd amicable pairs where the smaller term of the pair is less than 10^20

Cf. A066873, A259180, A262623, A262622, A262625, A005408.

A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

Original entry on oeis.org

67212, 1296000, 20528640, 37739520, 75479040, 321408000, 348364800, 556839360, 579156480, 638668800, 661893120, 761177088, 796340160, 883872000, 1181174400, 1282417920, 2068416000, 2395008000, 2682408960, 3155023872, 3599769600, 4049740800, 4606156800, 4716601344

Offset: 1

Zoltan Galantai, Dec 26 2022

nonn

From Michel Marcus, Dec 31 2022: (Start)
In other words, numbers k that can be expressed as a sum k = x+y = z+t where either (x,y) and (z,t), or (x,z) and (y,t), are 2 amicable pairs.
Note that there is currently a single instance of the case (x,z) and (y,t), and this corresponds to the value 64 that appears twice in A066539.
The other terms correspond to values appearing at least twice in A180164.
There are instances where it can appear 3 times, and the least instance is 64795852800 for the 3 amicable pairs [29912035725, 34883817075], [31695652275, 33100200525], [32129958525, 32665894275].
There are instances where it can appear 6 times, and the least instance is 4169926656000 for the 6 amicable pairs [1953433861918, 2216492794082], [1968039941816, 2201886714184], [1981957651366, 2187969004634], [1993501042130, 2176425613870], [2046897812505, 2123028843495], [2068113162038, 2101813493962]. (End)

			67212 is a term because 67212 = 220 + 66992 = 284 + 66928 where (220, 284) and (66928, 66992) are two amicable pairs.
1296000 is a term because 1296000 = 609928 + 686072 = 643336 + 652664 where (609928, 686072) and (643336, 652664) are two amicable pairs.

Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc, 1996, pp. 113-121.

Michel Marcus, Table of n, a(n) for n = 1..233
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025, A063990, A259180, A259933, A036471, A180164, A001065, A066539.

More terms from Amiram Eldar, Dec 31 2022

A297575 Numbers whose sum of divisors is divisible by 10.

Original entry on oeis.org

19, 24, 27, 29, 38, 40, 54, 56, 57, 58, 59, 76, 79, 87, 88, 89, 95, 104, 108, 109, 114, 116, 118, 120, 133, 135, 136, 139, 145, 149, 152, 158, 168, 171, 174, 177, 178, 179, 184, 189, 190, 199, 203, 209, 216, 218, 228, 229, 232, 236, 237, 239, 247, 248

Offset: 1

Zoltan Galantai, Jan 01 2018

nonn

Of the first 10^2, 10^6, and 10^10 positive integers, 17%, 40.8%, and 48.4%, respectively, are in the sequence (See Robert G. Wilson's table below). So the conjecture here is that if this trend continues then as the numbers approach infinity, the percentage of those numbers whose sum of divisors is divisible by 10 approaches 100%.
From David A. Corneth, Jan 01 2018: (Start)
If a(n) and m > 0 are coprime then a(n) * m is in the sequence.
A030433 is a subsequence.
Let p(n, d) be a prime ending in base-10 digit d and m != n gives p(m, d) != p(n, d). For p(n, 2) and p(n, 5) we must have n = 1 (one unique value). Then we could describe families of terms as factorizations in terms of these primes. 24 would give the family p(1, 2)^3 * p(1, 3).
For d coprime to 10 we could replace p(n, d) with p(n, 10 - d) and give another family of terms. Constructing terms could then be done by selecting primes for p(n, 1), p(n, 3), p(n, 7) and p(n, 9) from A030430, A030431, A030432 and A030433 respectively. (End)
From Robert G. Wilson v, Jan 03 2017: (Start)
The following array has for n powers of 10 and k is the number of integers <= 10^n which are == k (mod 10).
Array begins:
========================================================================================
n\k|         0     1          2     3          4     5          6     7          8     9
---|------------------------------------------------------------------------------------
_1 |         0     1          1     2          1     1          1     1          2     0
_2 |        17     6         16     4         21     2         12     4         17     1
_3 |       274    17        178    16        177     7        145     7        173     6
_4 |      3352    47       1690    44       1727    23       1498    27       1563    29
_5 |     37709   130      16012   144      16069    77      14649    87      15022   101
_6 |    408270   406     151936   424     152079   249     141527   293     144481   335
_7 |   4327266  1255    1451931  1341    1452017   787    1368110   923    1395278  1092
_8 |  45278675  3876   13963299  4206   13963783  2494   13268616  2998   13508556  3497
_9 | 469680154 11975  134976927 13185  134999718  7903  129084019  9684  131205200 11235
10 |4842279472 37545 1310133910 41349 1310381140 25017 1259051325 31136 1277983443 35663
  ... (End)
From Robert Israel, Jan 07 2018: (Start)
A number with prime factorization Product_j (p_j)^(e_j) is in the sequence if and only if
1) e_j is odd for some odd p_j, and
2) For some j, either p_j == 1 (mod 5) and e_j == 4 (mod 5), or p_j == 2 or 3 (mod 5) and e_j == 3 (mod 4), or p_j == 4 (mod 5) and e_j is odd.
By the strong form of Dirichlet's theorem, the product of 1-1/p for primes == 4 (mod 5) is 0, which implies that the asymptotic density of the sequence is 1.
(End)

			19 is in the sequence, since sigma(19) = 20, which is divisible by 10.
20 is not in the sequence as sigma(20) = 42, which isn't divisible by 10.

I. Niven, H. S. Zuckermann and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley and Sons, 1991, pages 4-20.

Zoltan Galantai, Table of n, a(n) for n = 1..1591
Zoltan Galantai, List of numbers up to 3 million whose sum of divisors is divisible by 10

Cf. A000203, A008592, A030430, A030431, A030432, A030433, A291422, A292217.

select(n -> numtheory:-sigma(n) mod 10 = 0, [$1..1000]); # Robert Israel, Jan 02 2018

Select[Range@250, Mod[DivisorSigma[1, #], 10] == 0 &] (* Robert G. Wilson v, Jan 03 2018 *)

is(n) = sigma(n) % 10 == 0 \\ David A. Corneth, Jan 01 2018

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

A291422 List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.

Original entry on oeis.org

6232, 6368, 10744, 10856, 12285, 14595, 66928, 66992, 67095, 71145, 79750, 88730, 100485, 124155, 122265, 139815, 122368, 123152, 141664, 153176, 142310, 168730, 176272, 180848, 185368, 203432, 356408, 399592, 437456, 455344, 522405, 525915, 600392, 669688, 609928, 686072

Offset: 1

Zoltan Galantai, Aug 22 2017

The sequence lists those amicable pairs (m,n) in increasing order where the sum of the amicable pair is divisible by ten.
Up to the first 5001 amicable pairs, 88.1% of the sums satisfy this condition (up to the first 100 amicable pairs: 74%; up to the first 1000: 82.5%; up to 2000: 85.25%).  So the conjecture here is that as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100%. [corrected by Paul Zimmermann, Feb 05-06 2019]
Among the 1947667 pairs up to 19 digits from Sergei Chernykh's database, there are 1872573 pairs with m+n divisible by ten, thus about 96.14%. - Paul Zimmermann, Feb 07 2019

			The sum of 6232 and 6368 is divisible by ten, thus the (6232, 6368) amicable pair belongs to the sequence. On the other hand, the (220, 284) amicable pair does not qualify since its sum is 504.

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 55-58.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 67-69.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sergei Chernykh, Amicable pairs list.
Zoltan Galantai, List of the amicable pairs where the sum divisible by ten smaller and larger amicable numbers; sums (the first 4406 pairs).
Zoltan Galantai, List of the first 5001 amicable pairs with their sums denoting whether the sum is divisible by ten or not.
Eric Weisstein's World of Mathematics, Amicable Pair.
Eric Weisstein's World of Mathematics, Sociable Numbers.

Cf. A002025, A002046, A063990, A180164, A259180, A259933, A291550.

lista(nn) = {for (n=1, nn, spd = sigma(n)-n; if ((spd > n) && (sigma(spd)-spd == n) && !((n + spd) % 10), print1(n, ", ", spd, ", ")););} \\ Michel Marcus, Aug 26 2017

A273259 Twin amicable numbers (m,n) where there is not any part of another amicable pair between m and n.

Original entry on oeis.org

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 66928, 66992, 122368, 123152, 196724, 202444, 437456, 455344, 469028, 486178, 503056, 514736, 522405, 525915, 643336, 652664, 802725, 863835, 998104, 1043096, 1077890, 1099390

Offset: 1

Zoltan Galantai, May 18 2016

nonn

The sequence lists the twin amicable number pairs in increasing order. So opposite to the list of the amicable numbers, the pairs (m, n) are necessarily adjacent to each other in this list.

First differs from A259933 at a(19). - Omar E. Pol, May 19 2016

			The (63020, 76084) amicable pair does not qualify since it overlaps with the (66928, 66992) pair.
On the other hand, 66928 and 66992 are contiguous, thus the (66928, 66992) amicable pair belongs to the sequence.

Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc, 1996, pp. 113-121.

Zoltan Galantai, Table of n, a(n) for n = 1..150
Bela Galantai, Software source code written in Clipper
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025 (smaller), A002046 (and larger of an amicable pair).

Cf. A063990 (amicable numbers), A259180 (amicable pairs), A259933.

cp's OEIS Frontend

User: Zoltan Galantai

A360694 Numbers whose divisors can be partitioned into two disjoint sets where the sum of both sets is prime.

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A360140 Odd amicable pairs with only one member divisible by 3.

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A360054 Number of odd amicable pairs where the smaller term of the pair is less than 10^n.

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A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

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A297575 Numbers whose sum of divisors is divisible by 10.

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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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A291422 List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.

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A273259 Twin amicable numbers (m,n) where there is not any part of another amicable pair between m and n.

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