A110928
Pairs of distinct numbers m and n, m
6, 7, 24, 26, 30, 35, 40, 47, 66, 77, 78, 91, 102, 119, 114, 133, 120, 130, 120, 141, 130, 141, 136, 157, 138, 161, 150, 175, 168, 182, 174, 203, 186, 215, 186, 217, 215, 217, 222, 259, 230, 249, 246, 287, 258, 301, 264, 286, 280, 282, 280, 329, 282, 329, 318
Offset: 1
Examples
sigma_2(30)=1^1+2^2+3^2+5^2+6^2+10^2+15^2+30^2=1300 and sigma_2(35)=1^2+5^2+7^2+35^2=1300.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Maple
with(numtheory); sigmap := proc(p,n) convert(map(proc(z) z^p end, divisors(n)),`+`) end; SA2:=[]: for z from 1 to 1 do for m to 1500 do M:=sigmap(2,m); for n from m+1 to 1500 do N:=sigmap(2,n); if N=M then SA2:=[op(SA2),[m,n,N]] fi od od od; SA2; select(proc(z) z[1]<=1000 end, SA2); #just to shorten it a bit
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Mathematica
a[n_] := Module[{s = DivisorSigma[2, n], ans = {}}, kmax = Ceiling[Sqrt[s]]; Do[If[DivisorSigma[2, k] == s, AppendTo[ans, k]], {k, n + 1, kmax}]; ans]; s = {}; Do[v = a[n]; Do[s = Join[s, {n, v[[k]]}], {k, 1, Length[v]}], {n, 1, 400}]; s (* Amiram Eldar, Sep 08 2019 *)
Formula
sigma_2(m)=sigma_2(n), m
A110929
The common value of sigma_2 for square-amicable numbers, sigma_2(m)=sigma_2(n), m
50, 850, 1300, 2210, 6100, 8500, 14500, 18100, 22100, 22100, 22100, 24650, 26500, 32550, 42500, 42100, 48100, 48100, 48100, 68500, 68900, 84100, 92500, 103700, 110500, 110500, 110500, 140500, 158600, 174100, 201110, 186100, 221000, 224500
Offset: 1
Keywords
Examples
sigma_2(30)=1^1+2^2+3^2+5^2+6^2+10^2+15^2+30^2=1300 and sigma_2(35)=1^2+5^2+7^2+35^2=1300.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Maple
with(numtheory); sigmap := proc(p,n) convert(map(proc(z) z^p end, divisors(n)),`+`) end; SA2:=[]: for z from 1 to 1 do for m to 1500 do M:=sigmap(2,m); for n from m+1 to 1500 do N:=sigmap(2,n); if N=M then SA2:=[op(SA2),[m,n,N]] fi od od od; SA2; select(proc(z) z[1]<=1000 end, SA2); #just to shorten it a bit
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Mathematica
a[n_] := Module[{s = DivisorSigma[2, n], ans = {}}, kmax = Ceiling[Sqrt[s]]; Do[If[DivisorSigma[2, k] == s, AppendTo[ans, s]], {k, n + 1, kmax}]; ans]; s = {}; Do[v = a[n]; Do[AppendTo[s, v[[k]]], {k, 1, Length[v]}], {n, 1, 400}]; s (* Amiram Eldar, Sep 08 2019 *)
Formula
sigma_2(m)=sigma_2(n), m
A127652 Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.
25, 28, 36, 40, 50, 68, 70, 74, 94, 95, 98, 116, 119, 134, 142, 143, 154, 162, 170, 175, 182, 189, 190, 200, 220, 226, 242, 245, 262, 273
Offset: 1
Comments
Here the length of an aliquot sequence is defined to be the length of the transient part of its trajectory + the length of its terminal cycle.
Examples
a(5)=50 because the fifth integer whose unitary aliquot sequence is longer than its ordinary aliquot sequence is 50.
References
- Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
- Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.
Links
- Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
- Wolfgang Creyaufmueller, Aliquot Sequences.
Crossrefs
Programs
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Mathematica
UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;g[n_] := If[n > 0, sstar[n], 0];UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];s[n_]:=DivisorSigma[1,n]-n;h[n_] := If[n > 0, s[n], 0];OrdinaryTrajectory[n_] := Most[NestWhileList[h, n, UnsameQ, All]];Select[Range[275],Length[UnitaryTrajectory[ # ]]>Length[OrdinaryTrajectory[ # ]] &]
A135244 Largest m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.
0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 26, 55, 289, 77, 361, 91, 38, 85, 529, 143, 46, 133, 28, 187, 841, 221, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 437, 1849, 403, 86, 493, 2209, 551, 94, 589, 0, 667, 2809, 713, 106, 703, 68, 697, 3481
Offset: 2
Keywords
Comments
Previous name: Aliquot predecessors with the largest values.
Find each node's predecessors in aliquot sequences and choose the largest predecessor.
Climb the aliquot trees on shortest paths (see A135245 = Climb the aliquot trees on thickest branches).
The sequence starts at offset 2, since all primes satisfy sigma(n)-n = 1. - Michel Marcus, Nov 11 2014
Examples
a(25) = 143 since 25 has 3 predecessors (95,119,143), 143 being the largest. a(5) = 0 since it has no predecessors (see Untouchables - A005114).
Links
- Amiram Eldar, Table of n, a(n) for n = 2..10000 (terms 2..150 from Ophir Spector)
- Wolfgang Creyaufmueller, Aliquot sequences.
- J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
- J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
- J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
- Eric Weisstein's World of Mathematics, Aliquot sequence.
Crossrefs
Programs
-
Mathematica
seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]] = Max[s[[i]], n]], {n, 2, (max - 1)^2}]; Rest @ s]; seq[50] -
PARI
lista(nn) = {for (n=2, nn, k = (n-1)^2; while(k && (sigma(k)-k != n), k--); print1(k, ", "););} \\ Michel Marcus, Nov 11 2014
Extensions
a(1)=0 removed and offset set to 2 by Michel Marcus, Nov 11 2014
New name from Michel Marcus, Oct 31 2023
A180202 The product of the two numbers in an amicable pair, A002025(n) * A002046(n).
62480, 1432640, 7660880, 27931280, 39685376, 116636864, 179299575, 318523136, 4794813680, 4483640576, 4773473775, 6100571295, 7076217500, 12475715175, 17094480975, 15069863936, 21699524864, 24011966300, 30304399616
Offset: 1
Keywords
Comments
This sequence initially shares many terms with A180163 because small amicable pairs are sometimes consecutive terms in the sorted list of amicable numbers, A063990.
First differs from A180163 at a(9). - Omar E. Pol, Oct 25 2017
Examples
a(9) = A002025(9) * A002046(9) = 63020 * 76084 = 4794813680.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Programs
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Mathematica
s[n_] := DivisorSigma[1,n]-n; smallAmicableQ[n_] := Module[{b=s[n]}, n
Formula
A180330 Smallest amicable number of the form 2^n * p * q for which the larger member of the amicable pair has the same form, where p and q are distinct odd primes.
2620, 10744, 66928, 2082464, 7677248, 1750776704, 749380864, 7074650624, 25937232896, 161899964416, 3949032574976, 56691934109696, 162222327218176, 5469697508737024, 21547979005558784, 48336727662002176, 2961911925308653568, 5591728346540539904
Offset: 2
Keywords
Comments
That is, the amicable pair is (2^n pq, 2^n rs) for odd primes p, q, r, s. See A180331 for the numbers 2^n rs. It is easy to show that the four primes must satisfy the equation (p+1)(q+1) = (r+1)(s+1). These amicable pairs are a subset of the regular type (2,2) pairs, which are cataloged by Pedersen. These amicable pairs can be found by using Herman te Riele's method 2. Amicable pairs of this form are known for 1 < n < 49. Do they exist for all n?
Links
- Sergei Chernykh, Amicable numbers list.
- Jan Munch Pedersen, Regular type (2,2) amicable pairs.
- Herman J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp. 42 (1984), 219-223.
Extensions
a(18)-a(19) from Chernykh's database added by Amiram Eldar, Jul 26 2025
A183019 Conjectured list of multisociable numbers.
6, 28, 120, 220, 284, 496, 672, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 12496, 14264, 14288, 14536, 14595, 15472, 17296, 18416, 30240, 32760, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633
Offset: 1
Keywords
A192290 Anti-amicable numbers.
14, 16, 92, 114, 5566, 6596, 1077378, 1529394, 3098834, 3978336, 70774930, 92974314
Offset: 1
Comments
Like A063990 but using anti-divisors. sigma*(a)=b and sigma*(b)=a, where sigma*(n) is the sum of the anti-divisors of n. Anti-perfect numbers A073930 are not included in the sequence.
There are also chains of 3 or more anti-sociable numbers.
With 3 numbers the first chain is: 1494, 2056, 1856.
sigma*(1494) = 4+7+12+29+36+49+61+103+332+427+996 = 2056.
sigma*(2056) = 3+9+16+1371+457 = 1856.
sigma*(1856) = 3+47+79+128+1237 = 1494.
With 4 numbers the first chain is: 46, 58, 96, 64.
sigma*(46) = 3+4+7+13+31 = 58.
sigma*(58) = 3+4+5+9+13+23+39 = 96.
sigma*(96) = 64.
sigma*(64) = 3+43 = 46.
No other pairs with the larger term < 2147000000. - Jud McCranie Sep 24 2019
Examples
sigma*(14) = 3+4+9 = 16; sigma*(16) = 3+11 = 14. sigma*(92) = 3+5+8+37+61= 114; sigma*(114) = 4+12+76 = 92. sigma*(5566) = 3+4+9+44+92+484+1012+1237+3711= 6596; sigma*(6596) = 3+8+79+136+776+167+4397 = 5566.
Programs
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Maple
with(numtheory); A192290 := proc(q) local a,b,c,k,n; for n from 1 to q do a:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od; b:=a; c:=0; for k from 2 to b-1 do if abs((b mod k)-k/2)<1 then c:=c+k; fi; od; if n=c and not a=c then print(n); fi; od; end: A192290(1000000000);
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Python
from sympy import divisors def sigma_s(n): return sum([2*d for d in divisors(n) if n > 2*d and n % (2*d)] + [d for d in divisors(2*n-1) if n > d >=2 and n % d] + [d for d in divisors(2*n+1) if n > d >=2 and n % d]) A192290 = [n for n in range(1,10**4) if sigma_s(n) != n and sigma_s(sigma_s(n)) == n] # Chai Wah Wu, Aug 14 2014
Extensions
a(7)-a(12) from Donovan Johnson, Sep 12 2011
A212327 Numbers k such that A001065(x)*x = k has at least two solutions.
36, 62480, 141440, 1245335, 1432640, 2286080, 6680960, 7660880, 27931280, 39685376, 116636864, 179299575, 318047135, 318523136, 358491735, 533718135, 709131500, 1119849500, 1122571695, 1814416175, 2081125376, 3565970135, 3991520000, 4141021500, 4483640576
Offset: 1
Keywords
Comments
Products of pairs of amicable numbers are members of this sequence.
Examples
For k = 36, A001065(6)*6 = 36, A001065(9)*9 = 36, therefore 36 is a term.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..51 (terms below 10^11)
- Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq., Vol. 26 (2023), Article 23.9.1.
Programs
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Mathematica
q[k_] := DivisorSum[k, 1 &, # * (DivisorSigma[1, #] - #) == k &] > 1; Select[Range[23*10^5], q] (* Amiram Eldar, Jul 01 2025 *)
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PARI
isok(k) = {my(d = divisors(k, 1), c = 0); for(i = 1, #d, if(d[i][1] * (sigma(d[i][2]) - d[i][1]) == k, c++; if(c == 2, break))); c == 2;} \\ Amiram Eldar, Jul 01 2025
Extensions
a(9)-a(25) from Donovan Johnson, May 21 2012
A259953 The sum (in nondecreasing order) of the two numbers in an amicable pair.
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
Keywords
Comments
Examples
------------------------------------------
A m i c a b l e p a i r Sum
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n A260086(n) + A260087(n) = a(n)
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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
...
Comments