A063990 -id:A063990 - cp's OEIS frontend

A127655 Numbers whose unitary aliquot sequences end in a unitary amicable pair, but which are not unitary amicable numbers themselves.

102, 388, 436, 484, 812, 866, 1020, 1036, 1040, 1116, 1196, 1380, 1500, 1524, 1532, 1552, 1618, 1644, 1716, 1724, 1726, 1744, 1916, 2020, 2066, 2068, 2324, 2368, 2386, 2486, 2592, 2684, 2880, 2924, 3032, 3098, 3120, 3124, 3136, 3276, 3400, 3442, 3444, 3446, 3482

Offset: 1

Ant King, Jan 25 2007

nonn

			a(5)=812 because the fifth non-unitary amicable number whose unitary aliquot sequence ends in a unitary amicable pair is 812.

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.

Amiram Eldar, Table of n, a(n) for n = 1..1000
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]

Cf. A063991, A097037, A127654, A098185, A097032, A097010, A127653, A097036.

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]];UnitaryAmicableNumberQ[k_]:=If[Nest[sstar,k,2]?k && !sstar[k]?k,True,False];Select[Range[2500],!UnitaryAmicableNumberQ[ # ] && UnitaryAmicableNumberQ[Last[UnitaryTrajectory[ # ]]] &]

More terms from Amiram Eldar, Apr 06 2019

A127656 Lengths of the exponential aliquot sequences.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 3, 4, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4

Offset: 1

Ant King, Jan 25 2007

nonn

The exponential aliquot sequence is defined by the map x -> A051377(x)-x starting at n.

The length of an exponential aliquot sequence is defined according to the length of its transient part + the length of its terminal cycle.

			a(4)=3 because the exponential aliquot sequence generated by 4 is <4,2,0> and it has length 3.
From _R. J. Mathar_, Oct 05 2017: (Start)
The aliquot sequnence may enter a cycle (see A054979)
36 -> 36 -> ..
180 -> 180 -> ..
252 -> 252 -> ..
396 -> 396 -> ..
468 -> 468 -> ..
612 -> 612 -> ..
684 -> 684 -> ..
828 -> 828 -> ..
900 -> 1260 -> 1260 -> ..
1044 -> 1044 -> ..
1116 -> 1116 -> ..
1260 -> 1260 -> ..
1332 -> 1332 -> ..
1352 -> 468 -> 468 -> ..
1476 -> 1476 -> ..
1548 -> 1548 -> ..
1692 -> 1692 -> ..
1728 -> 612 -> 612 -> ..
1800 -> 1800 -> ..
1908 -> 1908 -> ..
1980 -> 1980 -> ..
2124 -> 2124 -> ..
2196 -> 2196 -> ..
2340 -> 2340 -> ..
2412 -> 2412 -> ..
2556 -> 2556 -> ..
2628 -> 2628 -> ..
2700 -> 2700 -> ..
2772 -> 2772 -> ..
2844 -> 2844 -> ..
2880 -> 1800 -> 1800 -> ..
(End)

Hans Havermann, Table of n, a(n) for n = 1..10000
Hagis, Peter Jr., Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
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]

Cf. A127657, A127658, A127659, A127660.

Cf. also A127661.

A127656 := proc(n)
    local trac,x;
    x := n ;
    trac := [x] ;
    while true do
        x := A051377(x)-trac[-1] ;
        if x = 0 then
            return 1+nops(trac) ;
        elif x in trac then
            return nops(trac) ;
        end if;
        trac := [op(trac),x] ;
    end do:
end proc: # R. J. Mathar, Oct 05 2017

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n]; divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];se[n_]:=Plus@@ExponentialDivisors[n]-n;g[n_] := If[n > 0, se[n], 0];eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];Length[eTrajectory[ # ]] &/@Range[100]
(* Second program: *)
f[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}]&) /@ FactorInteger[n];
a[n_] := Length[FixedPointList[f[#]-#&, n]]-1;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 04 2023 *)

A127657 Integers whose exponential aliquot sequences end in an e-perfect number.

Original entry on oeis.org

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1352, 1476, 1548, 1692, 1728, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2880, 2916, 2988, 3000, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3750, 3852, 3924, 4068, 4140

Offset: 1

Ant King, Jan 25 2007

nonn

			a(5) = 468 because the fifth integer whose exponential aliquot sequences ends in an e-perfect number is 468.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some results concerning exponential divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
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]

Cf. A127656, A127658, A127659, A127660, A054979.

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n]; divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];se[n_]:=Plus@@ExponentialDivisors[n]-n;g[n_] := If[n > 0, se[n], 0];eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];ExponentialPerfectNumberQ[0]=False;ExponentialPerfectNumberQ[k_Integer] :=If[se[k]==k,True,False];Select[Range[5000],ExponentialPerfectNumberQ[Last[eTrajectory[ # ]]] &]
f[p_, e_] := DivisorSum[e, p^# &]; s[0] = s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] == v[[-2]] > 0]; Select[Range[4000], q] (* Amiram Eldar, Mar 11 2023 *)

A127658 Exponential aspiring numbers.

Original entry on oeis.org

900, 1352, 1728, 2880, 2916, 3000, 3750, 4356, 5292, 6480, 6760, 8100, 8640, 9464, 9900, 10404, 10648, 11700, 12000, 12096, 13500, 14580, 14872, 15300, 15552, 15876, 16000, 16200, 16224, 17100, 17836, 18252, 19008, 19044, 20160, 20412, 20700, 21780, 22464, 22500

Offset: 1

Ant King, Jan 25 2007

nonn

Exponential aspiring numbers are those integers whose exponential aliquot sequences end in an e-perfect number, but that are not e-perfect numbers themselves.

			a(5) = 2916 because the fifth non-e-perfect number whose exponential aliquot sequence ends in an e-perfect number is 2916.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some results concerning exponential divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
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]

Cf. A127656, A127657, A127659, A127660, A054979.

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n]; divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];se[n_]:=Plus@@ExponentialDivisors[n]-n;g[n_] := If[n > 0, se[n], 0];eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];Select[Range[25000],ExponentialPerfectNumberQ[Last[eTrajectory[ # ]]] && !ExponentialPerfectNumberQ[ # ]&]
f[p_, e_] := DivisorSum[e, p^# &]; s[0] = s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[32000], q] (* Amiram Eldar, Mar 11 2023 *)

A127659 Exponential amicable numbers.

Original entry on oeis.org

90972, 100548, 454860, 502740, 937692, 968436, 1000692, 1106028, 1182636, 1307124, 1546524, 1709316, 2092356, 2312604, 2638188, 2820132, 2915892, 3116988, 3365964, 3720276, 3729852, 3911796, 4122468, 4275684, 4323564, 4548600, 4688460, 4725756, 4821516, 4842180

Offset: 1

Ant King, Jan 25 2007

nonn

Union of A126165 and A126166. The first 10 terms of this sequence are the same as the first 10 terms of A127660.

			a(5)=937692 because the fifth non-e-perfect integer that satisfies A126164(A126164(n))=n is 937692.

Hagis, Peter Jr.; Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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]

Cf. A126164, A126165, A126166, A127656, A127657, A127658, A127660, A051377.

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n];divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];se[n_]:=Plus@@ExponentialDivisors[n]-n;g[n_] := If[n > 0, se[n], 0];eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];ExponentialAmicableNumberQ[k_]:=If[Nest[se,k,2]==k && !se[k]==k,True,False];Select[Range[5 10^6],ExponentialAmicableNumberQ[ # ] &]
fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m != n && esigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, May 09 2019 *)

Non-e-perfect numbers for which A126164(A126164(n))=n.

Link corrected by Andrew Lelechenko, Dec 04 2011

More terms from Amiram Eldar, May 09 2019

A127662 Integers whose infinitary aliquot sequences end in an infinitary perfect number (A007357).

Original entry on oeis.org

6, 30, 42, 54, 60, 66, 72, 78, 90, 100, 140, 148, 152, 192, 194, 196, 208, 220, 238, 244, 252, 268, 274, 292, 296, 298, 300, 336, 348, 350, 360, 364, 372, 374, 380, 382, 386, 400, 416, 420, 424, 476, 482, 492, 516, 520, 532, 540, 542, 544, 550, 572, 576, 578, 586

Offset: 1

Ant King, Jan 26 2007

			a(5) = 60 because the fifth number whose infinitary aliquot sequence ends in an infinitary perfect number is 60.
6 -> 6 ...
30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
54 -> 66 -> 78 -> 90 -> 90 -> ..
60 -> 60 -> ..
66 -> 78 -> 90 -> 90 -> ..
72 -> 78 -> 90 -> 90 -> ..
78 -> 90 -> 90 -> ..
90 -> 90 -> ..
100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
102 -> 114 -> 126 -> 114 -> ..  cycle but not in the sequence
114 -> 126 -> 114 -> .. cycle but not in the sequence
126 -> 114 -> 126 -> ..
140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
152 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
192 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
196 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
208 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
210 -> 366 -> 378 -> 582 -> 594 -> 846 -> 594 -> ..
220 -> 140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
238 -> 194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
244 -> 66 -> 78 -> 90 -> 90 -> ..
246 -> 258 -> 270 -> 450 -> 330 -> 534 -> 546 -> 798 -> 1122 -> 1470 -> 2130 -> 3054 -> 3066 -> 4038 -> 4050 -> 2346 -> 2838 -> 3498 -> 4278 -> 4938 -> 4950 -> 4410 -> 4590 -> 8370 -> 14670 -> 14850 -> 22590 -> 22770 -> 29070 -> 35730 -> 35910 -> 79290 -> 79470 -> 79650 -> 107550 -> 79650 -> ..

Amiram Eldar, Table of n, a(n) for n = 1..75
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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]

Cf. A007357, A126168, A127661 - A127667.

isA007357 := proc(n)
    A049417(n) = 2*n ;
    simplify(%) ;
end proc:
isA127662 := proc(n)
    local trac,x;
    x := n ;
    trac := [x] ;
    while true do
        x := A049417(x)-trac[-1] ;
        if x = 0 then
            return false ;
        elif x in trac then
            return isA007357(x) ;
        end if;
        trac := [op(trac),x] ;
    end do:
end proc:
for n from 1 do
    if isA127662(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Oct 05 2017

ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],?(And@@#&),{1}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryPerfectNumberQ[0]=False;InfinitaryPerfectNumberQ[k_Integer] :=If[properinfinitarydivisorsum[k]==k,True,False];Select[Range[500],InfinitaryPerfectNumberQ[Last[iTrajectory[ # ]]] &]
s[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[3200], q] (* Amiram Eldar, Mar 11 2023 *)

More terms from Amiram Eldar, Mar 11 2023

A275702 Numbers n whose deficiency is 26: 2n - sigma(n) = 26.

Original entry on oeis.org

58, 75, 328, 850, 1210, 2848, 35968, 537088, 549768921088, 8796145451008

Offset: 1

Timothy L. Tiffin, Aug 05 2016

Any term x = a(m) can be combined with any term y = A275701(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2.  Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one.  So far, these two sequences have produced only one amicable pair: (1210,1184) = (a(5),A275701(2)) = (A063990(4),A063990(3)).  If more are ever found, then they will also exhibit x-y = 26.
Notice that:
a(1) =     58 =   29*  2 = (4^1+25)*(4^1)/2
a(3) =    328 =   41*  8 = (4^2+25)*(4^2)/2
a(6) =   2848 =   89* 32 = (4^3+25)*(4^3)/2
a(7) =  35968 =  281*128 = (4^4+25)*(4^4)/2
a(8) = 537088 = 1049*512 = (4^5+25)*(4^5)/2.
If p = 4^k+25 is prime and n = p*(p-25)/2, then it is not hard to show that 2*n - sigma(n) = 26. The values of k in A204388 will guarantee that p is prime (A104072). Similarly, if q = 2*4^k+25 is prime and n = q*(q-25)/2, then 2*n - sigma(n) = 26. However, q will never be prime since it will always be divisible by 3: 2*4^k+25 == (2*1^k+25) mod 3 == 27 mod 3 == 0 mod 3. So, the following values will be in this sequence and provide upper bounds for the next seven terms:
(4^10+25)*(4^10)/2 = 549768921088 >= a(9)
(4^11+25)*(4^11)/2 = 8796145451008 >= a(10)
(4^17+25)*(4^17)/2 = 147573952804424777728 >= a(11)
(4^35+25)*(4^35)/2 = 696898287454081973187748591279228938354688 >= a(12)
(4^46+25)*(4^46)/2 = 12259964326927110866866776279099475433218926722425028608 >= a(13)
(4^56+25)*(4^56)/2 = 13479973333575319897333507543509880240529303896615642871755920375808 >= a(14)
(4^59+25)*(4^59)/2 = 55213970774324510299478046898216207773446358605225195265697257166471168 >= a(15).
The rightmost digit of n = p*(p-25)/2 will always be 8. [Proof: If k is odd, then 4^k+25 == 9 mod 10 and (4^k)/2 == 2 mod 10, which implies that p*(p-25)/2 == 8 mod 10. If k is even, then 4^k+25 == 1 mod 10 and (4^k)/2 == 8 mod 10, which implies that p*(p-25)/2 == 8 mod 10.]
a(10) > 2.3*10^12. - Giovanni Resta, Aug 07 2016
a(11) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018

			a(1) = 58, since 2*58-sigma(58) = 116-90 = 26.
a(2) = 75, since 2*75-sigma(75) = 150-124 = 26.
a(3) = 328, since 2*328-sigma(328) = 656-630 = 26.

D. Alpern, Factorization using the Elliptic Curve Method.

Cf. A033879, A063990, A104072, A204388, A275701 (abundance 26).

[n: n in [1..2*10^6] | (2*n-SumOfDivisors(n)) eq 26]; // Vincenzo Librandi, Aug 06 2016

Select[Range[10^6], 2 # - (DivisorSigma[1, #]) == 26 &] (* Vincenzo Librandi, Aug 06 2016 *)

is(n) = 2*n-sigma(n)==26 \\ Felix Fröhlich, Aug 06 2016

a(9) from Giovanni Resta, Aug 07 2016

a(10) from Hiroaki Yamanouchi, Aug 21 2018

A383239 Integers k such that there exists an integer 0

Original entry on oeis.org

1740, 7776, 22428, 55968, 106140, 143910, 198792, 246510, 309582, 326196, 411138, 421596, 428256, 590112, 639288, 697158, 870552, 941094, 958716, 1060956, 1087776, 1105884, 1269828, 1341660, 1361568, 1447620, 1495494, 1512810, 1626324, 1727940, 1819392

Offset: 1

S. I. Dimitrov, Apr 20 2025

S. I. Dimitrov introduced the notion of (alpha_1,...,alpha_k)-multiamicable k-tuples.

The asymptotic density of (alpha_1, alpha_2)-multiamicable pairs relative to the positive integers is 0.

			For k=2, alpha_1=1, alpha_2=2 we have (1560, 1740), (7380, 7776), (20664, 22428), (543456, 590112), (588744, 639288),

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Subsequence of A259302.

Cf. A000203, A005820, A063990, A259180, A125490, A036474, A259303, A292365.

isok(k) = my(s=sigma(k), m=s-2*k); m>0 && mMichel Marcus, Apr 28 2025

We say that the natural numbers n_1,..., n_k form an (alpha_1,...,alpha_k)-multiamicable k-tuple if sigma(n_1)=sigma(n_2)=...=sigma(n_k)=alpha_1n_1+alpha_2n_2+...+alpha_kn_k, where alpha_1,...,alpha_k are positive integers, where sigma(n) is the sum of the divisors of n.

More terms from Sean A. Irvine, May 04 2025

A110926 Smaller of the pair of distinct numbers m and n such that sigma_2(m)=sigma_2(n), where sigma_2(n) is the sum of the squares of all divisors of n.

Original entry on oeis.org

6, 24, 30, 40, 66, 78, 102, 114, 120, 120, 130, 136, 138, 150, 168, 174, 186, 186, 215, 222, 230, 246, 258, 264, 280, 280, 282, 318, 330, 354, 360, 366, 390, 402, 408, 408, 426, 430, 438, 440, 442, 456, 474, 498, 510, 520, 534, 552, 570, 582, 600, 606, 618

Offset: 1

Walter Kehowski, Sep 23 2005

nonn

There do not appear to be any pairs (m,n) such that sigma_k(m)=sigma_k(n) for k>2.

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001157, A002025, A002046, A063990.

Cf. A110927, A110928, A110929.

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

sigma_2(m)=sigma_2(n), m

A110927 Larger of the pair of distinct numbers m and n such that sigma_2(m)=sigma_2(n), where sigma_2(n) is the sum of the squares of all divisors of n.

Original entry on oeis.org

7, 26, 35, 47, 77, 91, 119, 133, 130, 141, 141, 157, 161, 175, 182, 203, 215, 217, 217, 259, 249, 287, 301, 286, 282, 329, 329, 371, 385, 413, 423, 427, 455, 469, 442, 471, 497, 434, 511, 517, 471, 494, 553, 581, 595, 611, 623, 598, 665, 679, 650, 707, 721

Offset: 1

Walter Kehowski, Sep 23 2005

nonn

There do not appear to be any pairs (m,n) such that sigma_k(m)=sigma_k(n) for k>2.

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001157, A002025, A002046, A063990.

Cf. A110926, A110928, A110929.

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

sigma_2(m)=sigma_2(n), m

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A127655 Numbers whose unitary aliquot sequences end in a unitary amicable pair, but which are not unitary amicable numbers themselves.

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A127656 Lengths of the exponential aliquot sequences.

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Maple

Mathematica

A127657 Integers whose exponential aliquot sequences end in an e-perfect number.

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Mathematica

A127658 Exponential aspiring numbers.

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Mathematica

A127659 Exponential amicable numbers.

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A127662 Integers whose infinitary aliquot sequences end in an infinitary perfect number (A007357).

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Maple

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A275702 Numbers n whose deficiency is 26: 2n - sigma(n) = 26.

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Magma

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PARI

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A383239 Integers k such that there exists an integer 0