A002827 -id:A002827 - cp's OEIS frontend

A129486 Odd unitary abundant numbers that are not odd, squarefree, ordinary abundant numbers.

195195, 333795, 416955, 1786785, 1996995, 2417415, 2807805, 3138135, 3318315, 3708705, 3798795, 4103715, 4339335, 4489485, 4789785, 4967655, 5120115, 5420415, 5552085, 5660655, 5731635, 6051045, 6111105, 6263565, 6342105, 6695535, 6771765, 6938295, 7000455, 7088235

Offset: 1

Ant King, Apr 17 2007

The first 50 members of A129485 and A112643 are the same. However, the sequences differ thereafter and this sequence contains those integers that are included in A129485 but are not included in A112643.

			The third integer which is an odd unitary abundant number but is not an ordinary, squarefree abundant number is 416955. Hence a(3)=416955.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein, Unitary Divisor.

Cf. A034683, A129485, A034460, A034448, A129487, A002827, A129468, A112643.

UnitaryDivisors[ n_Integer?Positive ] := Select[ Divisors[ n ], GCD[ #, n/# ] == 1 & ]; sstar[ n_ ] := Plus @@ UnitaryDivisors[ n ] - n; UnitaryAbundantNumberQ[ k_ ] := If[ sstar[ k ] > k, True, False ]; data1 = Select[ Range[ 1, 10^7, 2 ], UnitaryAbundantNumberQ[ # ] & ]; data2 = Select[ Range[ 1, 10^7, 2 ], DivisorSigma[ 1, # ] - 2 # > 0 && ! MoebiusMu[ # ] == 0 & ]; Complement[ data1, data2 ]
uaQ[n_] := Module[{f = FactorInteger[n]}, Max[f[[;;,2]]] > 1 && Times@@(1 + Power @@@ f) > 2n]; Select[Range[3, 2*10^6, 2], uaQ] (* Amiram Eldar, May 13 2019 *)

The complement of A129485 and A112643.

More terms from Amiram Eldar, May 13 2019

A258142 Consider the unitary aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Original entry on oeis.org

6, 21, 60, 85, 90, 261, 976, 2009, 87360, 97273, 4948133, 68353213

Offset: 1

Paolo P. Lava, May 22 2015

A002827 is a subset of this sequence.

No more terms below 10^8. - Amiram Eldar, Jan 12 2019

			Divisors of 85 are 1, 5, 17, 85. Unitary aliquot parts are 1, 5, 17.
We have:
1 + 5 + 17 = 23;
5 + 17 + 23 = 45;
17 + 23 + 45 = 85.
Divisors of 2009 are 1, 7, 41, 49, 287, 2009.
Unitary aliquot parts are 1, 41, 49. We have:
1 + 41 + 49 = 91;
41 + 49 + 91 = 181;
49 + 91 + 181 = 321;
91 + 181 + 321 = 593;
181 + 321 + 593 = 1095;
321 + 593 + 1095 = 2009.

Eric Weisstein's World of Mathematics, Unitary Divisor
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Eric Weisstein's World of Mathematics, Unitary Perfect Number
Wikipedia, Unitary divisor

Cf. A002827, A034444, A246544, A247012, A247013, A248134.

with(numtheory):P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 1 to q do if not isprime(n) then b:=sort([op(divisors(n))]); a:=[];
for k from 1 to nops(b)-1 do if gcd(b[k],n/b[k])=1 then a:=[op(a),b[k]]; fi; od;
a:=sort(a); b:=nops(a); if b>1 then for k from 1 to b do v[k]:=a[k]; od;
t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]

aQ[n_] := Module[{s = Most[Select[Divisors[n], GCD[#, n/#] == 1 &]]}, If[Length[s] == 1, False, While[Total[s] < n, AppendTo[s, Total[s]]; s = Rest[s]]; Total[s] == n]]; Select[Range[2, 10^8], aQ] (* Amiram Eldar, Jan 12 2019 *)

a(11)-a(12) from Amiram Eldar, Jan 12 2019

A319937 Unitary sociable numbers of order 10.

Original entry on oeis.org

525150234, 527787366, 528544218, 553128198, 612951066, 675192294, 735821562, 982674438, 998151162, 998151174, 5251502340, 5277873660, 5285442180, 5531281980, 6129510660, 6751922940, 7358215620, 9826744380, 9981511620, 9981511740

Offset: 1

Michel Marcus, Oct 02 2018

J. O. M. Pedersen, Known Unitary Sociable Numbers of order different from four [Via Internet Archive Wayback-Machine]
Eric Weisstein's World of Mathematics, Unitary Sociable Numbers
J. O. M. Pedersen, Order 10 cycles

Cf. A063919 (sum of proper unitary divisors).
Cf. A002827 (unitary perfect), A063991 (unitary amicable).
Cf. A097024 (order 5), A319917 (order 6), A097030 (order 14).

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok10(n) = iferr(f(f(f(f(f(f(f(f(f(f(n)))))))))) == n, E, 0);
isok5(n) = iferr(f(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) = isok10(n) && !isok1(n) && !isok2(n) && !isok5(n);

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

Original entry on oeis.org

6, 28, 264, 1104, 3360, 75840, 151062912, 606557952, 2171581440

Offset: 1

Amiram Eldar, Apr 10 2020

Since a recursive divisor is also a (1+e)-divisor (see A049599), then the first 6 terms and other terms of this sequence coincide with those of A049603.

			264 is a term since the sum of its recursive divisors is 1 + 2 + 3 + 6 + 8 + 11 + 22 + 24 + 33 + 66 + 88 + 264 = 528 = 2 * 264.

Cf. A049603, A282446, A333926.

Analogous sequences: A000396, A002827 (unitary), A007357 (infinitary), A054979 (exponential), A064591 (nonunitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == 2*# &]

A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

Original entry on oeis.org

6, 15, 20, 24, 28, 30, 45, 60, 72, 90, 91, 96, 100, 112, 153, 216, 220, 240, 264, 272, 325, 352, 360, 364, 378, 496, 703, 765, 780, 816, 832, 1056, 1125, 1170, 1225, 1360, 1431, 1512, 1656, 1760, 1891, 1900, 1984, 2275, 2448, 2520, 2701, 2912, 3024, 3168, 3321

Offset: 1

Amiram Eldar, May 29 2020

nonn

Since the unitary divisors of a power of prime (A000961), p^e, are {1, p^e}, they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 3, 5, 6, 6, 7, 5, 9, 7, 12, 7, 13, 8, 10, 14, 17, ...
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m) | m * (2^omega(m)-1), or A063919(m) | (m * A309307(m)), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) = A034444(m) is the number of the unitary divisors of m.
The squarefree terms of A335267 are also terms of this sequence.
The terms with 2 distinct prime divisors are of the form p^e * (2*p^e - 1), when the second factor is also a prime power. The least term which both of its 2 prime divisors are nonunitary (with multiplicity larger than 1) is 1225 = 5^2 * 7^2 = 5^2 * (2 * 5^2 - 1).
The unitary perfect numbers (A002827) are terms of this sequence: if m is a unitary perfect number then usigma(m)-m = m.

			6 is a term since its unitary divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.

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

The unitary version of A335267.
A002827 is subsequence.
Cf. A006086, A000961, A024619, A034444, A034448, A063919, A077610, A309307, A335269, A335270.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3000], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - #] &]

A098188 Irregular triangle with 4 columns which contains in each row the members of a 4-cycle under the map x->A063919(x), iteration of summing the proper-unitary divisors.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Sep 02 2004

Initial values attracted by this sequence are in A098187.

The iteration of this function also contains 2-cycles like 114->126->114... or 1140 -> 1260 ->1140,... or 3-cycles like 30->42->54->30->....

			The first line represents the 4-cycle  280500->263820->263940->280380->280500->...,
The second line represents the 4-cycle 420750->395730->395910->420570->420750->..

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. A098185, A098186, A002827, A097024, A097030, A098187.

Cf. A319902 (where the terms are entered by increasing value).

More terms from Michel Marcus, Oct 05 2018

A335141 Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).

Original entry on oeis.org

840, 2940, 7260, 9240, 10140, 10920, 13860, 14280, 15960, 16380, 17160, 18480, 19320, 20580, 21420, 21840, 22440, 23100, 23940, 24024, 24360, 25080, 26040, 26520, 27300, 28560, 29640, 30360, 30870, 31080, 31920, 32340, 34440, 34650, 35700, 35880, 36120, 36960

Offset: 1

Amiram Eldar, May 25 2020

nonn

All the terms are nonsquarefree (since squarefree numbers do not have nonunitary divisors).

All the terms are either 3-abundant numbers (A068403) or 3-perfect numbers (A005820). None of the 6 known 3-perfect numbers are terms of this sequence. If there is a term that is 3-perfect, it is also a unitary perfect (A002827) and a nonunitary perfect (A064591).

			840 is a term since its aliquot unitary divisors are {1, 3, 5, 7, 8, 15, 21, 24, 35, 40, 56, 105, 120, 168, 280} and 1 + 5 + 7 + 8 + 15 + 35 + 40 + 56 + 105 + 120 + 168 + 280 = 840, and its nonunitary divisors are {2, 4, 6, 10, 12, 14, 20, 28, 30, 42, 60, 70, 84, 140, 210, 420} and 70 + 140 + 210 + 420 = 840.

Amiram Eldar, Table of n, a(n) for n = 1..400

Intersection of A293188 and A327945.
Subsequence of A335140.
Cf. A002827, A005820, A064591, A068403.

pspQ[n_] := Module[{d = Divisors[n], ud, nd, x}, ud = Select[d, CoprimeQ[#, n/#] &]; nd = Complement[d, ud]; ud = Most[ud]; Plus @@ ud >= n && Plus @@ nd >= n && SeriesCoefficient[Series[Product[1 + x^ud[[i]], {i, Length[ud]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^nd[[i]], {i, Length[nd]}], {x, 0, n}], n] > 0]; Select[Range[10^4], pspQ]

A063829 usigma(n) = 2n + d(n), where d(n) is the number of divisors of n.

Original entry on oeis.org

150, 294, 726, 1014, 1428, 1734, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 20886, 22326, 26934, 30246, 31974, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206, 136806, 147894, 159414

Offset: 1

Jason Earls, Aug 20 2001

The sequence includes all numbers of the form 6 * p^2 with p a prime >= 5. All of the terms above are of this form, except for 1428. There are similar subsequences corresponding to each of the five known unitary perfect numbers (A002827), namely 60 * p^9 (p>=7), 90 * p^14 (p>=7), 87360 * p^1559 (p=11 or p>=17) and 146361946186458562560000 * p^3009086064688703999 (p>=17 and not equal to 19, 37, 79, 109, 157, or 313). It is not known if there are other terms in the sequence besides these and 1428. - Dean Hickerson

The term 33872160 was found later: it is not of the form a * p^e where a is a unitary perfect number and p is a prime not dividing a. - Jason Earls

Donovan Johnson, Table of n, a(n) for n = 1..5000

Cf. A002827, A034448.

usigma[n_] := Sum[d*Boole[GCD[d, n/d] == 1], {d, Divisors[n]}]; Reap[For[n = 1, n < 140000, n++, If[usigma[n] == 2 n + DivisorSigma[0, n], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 16 2013 *)

us(n)=sumdiv(n,d, if(gcd(d,n/d)==1,d));
for(n=1,10^8, if(us(n)==2*n+numdiv(n),print1(n, ", ")))

A098187 Initial seeds x which will enter a cycle of length 4 under the iteration of x -> A063919(x), the sum of proper unitary divisors.

Original entry on oeis.org

81570, 114270, 137046, 169998, 177906, 182082, 182094, 185190, 194574, 194586, 201642, 203442, 204420, 204540, 212466, 212874, 213870, 219306, 219318, 230874, 231438, 231834, 231846, 232626, 237678, 238134, 242634, 258882, 259338, 259350

Offset: 1

Labos Elemer, Sep 02 2004

nonn

The sequence is the attractor-basin of set of {C4} cycles belonging to this iteration.

The {C4} attractor-set is displayed separately in A098188.

			81570 is in the sequence because its track under the iterated map is 81570, 114270, 182082, 182094, 232626, 237678, 305682, 352878, 360978, 403662, 420738, [420750, 395730, 395910, 420570], 420750.., where the cycle is indicated by brackets. The 4 recurrent terms appear after 11 transients for this case.

Cf. A098185, A098186, A002827, A097024, A097030.

A334406 Unitary pseudoperfect numbers k such that there is a subset of unitary divisors of k whose sum is 2*k and for each d in this subset k/d is also in it.

Original entry on oeis.org

6, 60, 90, 210, 330, 546, 660, 714, 1770, 2310, 2730, 3198, 3486, 3570, 3990, 4290, 4620, 4830, 5460, 5610, 6006, 6090, 6270, 6510, 6630, 6930, 7140, 7410, 7590, 7770, 7854, 7980, 8190, 8580, 8610, 8778, 8970, 9030, 9240, 9570, 9660, 9690, 9870, 10374, 10626, 10710

Offset: 1

Amiram Eldar, Apr 27 2020

nonn

Includes all the unitary perfect numbers (A002827).

The squarefree terms of A334405 are also terms of this sequence. Terms that are not squarefree are 60, 90, 660, 4620, 5460, 6930, 7140, 7980, 8190, 8580, 9240, 9660, ...

			210 is a term since {1, 2, 3, 14, 15, 70, 105, 210} is a subset of its unitary divisors whose sum is 420 = 2 * 210, and for each divisor d in this subset 210/d is also in it: 1 * 210 = 2 * 105 = 3 * 70 = 14 * 15 = 210.

Amiram Eldar, Table of n, a(n) for n = 1..450

Subsequence of A293188 and A334405.
A002827 is a subsequence.
Cf. A077610.

seqQ[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &]}, nd = Length[d]; divpairs = d[[1 ;; nd/2]] + d[[-1 ;; nd/2 + 1 ;; -1]]; SeriesCoefficient[Series[Product[1 + x^divpairs[[i]], {i, Length[divpairs]}], {x, 0, 2*n}], 2*n] > 0]; Select[Range[2, 1000], seqQ]

cp's OEIS Frontend

A129486 Odd unitary abundant numbers that are not odd, squarefree, ordinary abundant numbers.

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A258142 Consider the unitary aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

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A319937 Unitary sociable numbers of order 10.

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PARI

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

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A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

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A098188 Irregular triangle with 4 columns which contains in each row the members of a 4-cycle under the map x->A063919(x), iteration of summing the proper-unitary divisors.

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A335141 Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).

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A063829 usigma(n) = 2n + d(n), where d(n) is the number of divisors of n.

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