A129487 -id:A129487 - cp's OEIS frontend

A034683 Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834

Offset: 1

Erich Friedman

nonn

If a term n in the sequence ends in neither 0 nor 5, then 10*n is also in the sequence. - Lekraj Beedassy, Jun 11 2004
The lower asymptotic density of this sequence is larger than 1/18 = 0.0555... which is the density of its subsequence of numbers of the form 6*m where gcd(m, 6) = 1 and m > 1. Numerically, based on counts of terms below 10^n (A302993), it seems that this sequence has an asymptotic density which equals to about 0.070034... - Amiram Eldar, Feb 13 2021
The asymptotic density of this sequence is in the interval (0.0674, 0.1055) (Wall, 1970). - Amiram Eldar, Apr 18 2024
All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, usigma(k)/k < 15/Pi^2 = 1.519817... (A082020; the record values are attained at the squares of primorials, A061742). - Amiram Eldar, Jul 20 2024

C. Sung, Mathematical Buds, "Unitary Divisors", Chap. V, pp. 42-67, Ed. H. D. Ruderman, Mu Alpha Theta OK 1978.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

Subsequence of A005101.
Cf. A034444, A034448, A129487, A302993.
Cf. A001694, A061742, A082020.

isA034683 := proc(n)
    is(A034448(n) > 2*n) ;
end proc:
for n from 1 do
    if isA034683(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
Select[Range[1000], usigma[#] > 2#&] (* Jean-François Alcover, Mar 23 2020, after Giovanni Resta in A034448 *)

is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]) > 2*n;} \\ Amiram Eldar, Apr 18 2024

A129485 Odd unitary abundant numbers.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035

Offset: 1

Ant King, Apr 17 2007

This sequence is different from A112643. The two sequences agree for the first 50 terms but differ thereafter. The exceptions, i.e. those odd unitary abundant numbers that are not squarefree ordinary abundant numbers, are in A129486.
22309287 is the smallest term not divisible by 5. 33426748355 is the smallest term not divisible by 3. - Donovan Johnson, May 15 2013
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 34, 137, 1714, 16918, 181744, 1752337, 17290556, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00017... . - Amiram Eldar, Sep 02 2022

			The third odd unitary abundant number is 21945. Hence a(3) = 21945.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A034683, A129486, A034460, A034448, A129487, A002827, A129468, A112643.

# see A034683 for the code of isA034683()
isA129485 := proc(n)
    type(n,'odd') and isA034683(n) ;
end proc:
for n from 1 do
    if isA129485(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;Select[Range[1,10^5,2],sstar[ # ]># &]

This sequence contains the odd members of A034683. i.e. odd numbers with a positive unitary abundance (A129468).

A302574 Primitive unitary abundant numbers (definition 2): unitary abundant numbers (A034683) having no unitary abundant proper unitary divisor.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 660, 678, 726, 750, 762, 780, 786, 822, 834, 840, 894, 906, 924, 942, 978, 990, 1002, 1014, 1020, 1038, 1074

Offset: 1

Amiram Eldar, Apr 10 2018

nonn

The unitary analog of A091191.

			70 is primitive unitary abundant since it is unitary abundant (usigma(70) = 144 > 2*70), and all of its unitary divisors are unitary deficient. 210 is unitary abundant since usigma(210) = 576 > 2*210, but is not in this sequence since 70 is one of its unitary divisors, and 70 is unitary abundant.

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

Cf. A034448, A034683, A091191, A129487, A302573.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; delta[n_] := usigma[n]-2n; udefQ[n_] := Module[{}, v=Most[Module[{d = Divisors[n]}, Select[ d, GCD[ #, n/# ] == 1 &]]]; u = Max[Map[delta,v]]; u<=0 ]; puaQ[n_] := delta[n] > 0 && udefQ[n]; Select[Range[10000],puaQ]

A129468 Unitary abundance of n.

Original entry on oeis.org

-1, -1, -2, -3, -4, 0, -6, -7, -8, -2, -10, -4, -12, -4, -6, -15, -16, -6, -18, -10, -10, -8, -22, -12, -24, -10, -26, -16, -28, 12, -30, -31, -18, -14, -22, -22, -36, -16, -22, -26, -40, 12, -42, -28, -30, -20, -46, -28, -48, -22, -30, -34, -52, -24

Offset: 1

Ant King, Apr 17 2007

The values of n which generate negative elements of this sequence are in A129487, the values of n which generate the zeros of this sequence are in A002827 and the values of n which generate positive elements of this sequence are in A034683

			As the unitary divisors of 12 are 1, 3, 4 and 12, which sum to 20, then a(12) = 20 - 2*12 = -4.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A034460, A034448, A129487, A002827, A034683.

Cf. A002117, A013661.

A129468 := proc(n)
    A034448(n)-2*n ;
end proc:
seq(A129468(n),n=1..40) ; # R. J. Mathar, Nov 10 2014

UnitaryDivisors[n_Integer?Positive] := Select[Divisors[n], GCD[ #,n/# ] == 1&]; sstar[n_] := Plus@@UnitaryDivisors[n] - n; sstar[ # ] - # &/@ Range[40]
a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - 2*n; a[1] = -1; Array[a, 100] (* Amiram Eldar, Apr 06 2024 *)

a(n) = {my(f = factor(n)); prod(i=1, #f~, 1 + f[i, 1]^f[i, 2]) - 2*n; } \\ Amiram Eldar, Apr 06 2024

a(n) = A034460(n) - n = A034448(n) - 2n.
From Amiram Eldar, Apr 06 2024: (Start)
a(A129487(n)) < 0.
a(A002827(n)) = 0.
a(A034683(n)) > 0.
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(2*zeta(3)) - 1 = -0.3157836111... . (End)

A097010 Numbers n such that zero is eventually reached when the map x -> A034460(x) is iterated, starting from x = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79

Offset: 1

Labos Elemer, Aug 31 2004

nonn

Numbers n for which A318880(n) = 0. - Antti Karttunen, Sep 23 2018

The sequence doesn't contain any numbers from attractor sets like A002827, A063991, A097024, A097030, etc, nor any number x such that the iteration of the map x -> A034460(x) would lead to such an attractor set (e.g., numbers in A097034 - A097037). - Antti Karttunen, Sep 24 2018, after the original author's example.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A002827, A034460, A063991, A097024, A097030, A097031, A097032, A097033, A097034, A097035, A097036, A097037.
Cf. A003062 (complement), A318880.
Differs from A129487 for the first time at n=51, as A129487(51) = 54, but that term is lacking here, as in this sequence a(51) = 55.

di[x_] :=Divisors[x];ta={{0}}; ud[x_] :=Part[di[x],Flatten[Position[GCD[di[x],Reverse[di[x]]],1]]]; asu[x_] :=Apply[Plus,ud[x]]-x;nsf[x_,ho_] :=NestList[asu,x,ho] Do[g=n;s=Last[NestList[asu,n,100]];If[Equal[s,0],Print[{n,s}]; ta=Append[ta,n]],{n,1,256}];ta = Delete[ta,1]

up_to = 10000;
A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A318880(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(1), mapput(visited, n, j)); n = A034460(n); if(!n,return(0))); };
A097010list(up_to) = { my(v = vector(up_to), k=0, n=1); while(kA318880(n), k++; v[k] = n); n++); (v); };
v097010 = A097010list(up_to);
A097010(n) = v097010[n]; \\ Antti Karttunen, Sep 24 2018

Edited by Antti Karttunen, Sep 24 2018

A302573 Primitive unitary abundant numbers (definition 1): unitary abundant numbers (A034683) all of whose proper unitary divisors are unitary deficient.

Original entry on oeis.org

70, 840, 924, 1092, 1386, 1428, 1430, 1596, 1638, 1870, 2002, 2090, 2142, 2210, 2394, 2470, 2530, 2970, 2990, 3190, 3230, 3410, 3510, 3770, 4030, 4070, 4510, 4730, 5170, 5390, 5830, 13860, 15015, 16380, 17160, 18480, 19635, 20020, 21420, 21840, 21945, 22440

Offset: 1

Amiram Eldar, Apr 10 2018

nonn

The unitary analog of A071395.

Prasad & Reddy proved that n is a primitive unitary abundant number if and only if 0 < usigma(n) - 2n < 2n/p^e, where p^e is the largest prime power that divides n.

			70 is primitive unitary abundant since it is unitary abundant (usigma(70) = 144 > 2*70), and all of its unitary divisors are unitary deficient. The smaller unitary abundant numbers, 30, 42, 66, are not primitive, since in each 6 is a unitary divisor, and 6 is not unitary deficient.

J. Sandor, D. S. Mitrinovic, and B. Crstici, Handbook of Number Theory, Vol. 1, Springer, 2006, p. 115.

Amiram Eldar, Table of n, a(n) for n = 1..10000
V. Siva Rama Prasad and D. Ram Reddy, On primitive unitary abundant numbers, Indian J. Pure Appl. Math., Vol. 21, No. 1 (1990) pp. 40-44.

Cf. A034448, A034683, A071395, A129487, A302574.

maxPower[n_]:=Max[Power @@@ FactorInteger[n]];usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; d[n_]:=usigma[n]-2n; punQ[n_] := d[n]>0 && d[n]< 2n/maxPower[n]; Select[Range[1000], punQ]

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

Original entry on oeis.org

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

A335252 Numbers k such that k and k+2 have the same unitary abundance (A129468).

Original entry on oeis.org

12, 63, 117, 323, 442, 1073, 1323, 1517, 3869, 5427, 6497, 12317, 18419, 35657, 69647, 79919, 126869, 133787, 151979, 154007, 163332, 181427, 184619, 333797, 404471, 439097, 485237, 581129, 621497, 825497, 1410119, 2696807, 3077909, 3751619, 5145341, 6220607

Offset: 1

Amiram Eldar, May 28 2020

nonn

Are 12, 442 and 163332 the only even terms?
Are there any unitary abundant numbers (A034683) in this sequence?
No further even terms up to 10^13. - Giovanni Resta, May 30 2020

			12 is a term since 12 and 14 have the same unitary abundance: A129468(12) = usigma(12) - 2*12 = 20 - 24 = -4, and A129468(14) = usigma(14) - 2*14 = 24 - 28 = -4.

Amiram Eldar, Table of n, a(n) for n = 1..300

The unitary version of A330901.

Cf. A034448, A034683, A129468, A129487, A335251.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); udef[n_] := 2*n - usigma[n]; Select[Range[10^5], udef[#] == udef[# + 2] &]

A335251 Numbers k such that k and k+1 have the same unitary abundance (A129468).

Original entry on oeis.org

1, 20, 35, 143, 208, 2623, 5183, 27796, 11177983, 69677008, 920158207, 1099508482048

Offset: 1

Amiram Eldar, May 28 2020

Are there any unitary abundant numbers (A034683) in this sequence?
a(12) > 10^11.
a(13) > 8*10^12. Also terms: 2^36 * 68719644673, 2^48 * 281474901625261, 2^64 * 18446632096776339457. - Giovanni Resta, May 29 2020

			1 is a term since 1 and 2 have the same unitary abundance: A129468(1) = usigma(1) - 2*1 = 1 - 2 = -1, and A129468(2) = usigma(2) - 2*2 = 3 - 4 = -1.

Cf. A034448, A034683, A129468, A129487, A331412, A335252.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); udef[n_] := 2*n - usigma[n]; Select[Range[30000], udef[#] == udef[# + 1] &]

a(12) from Giovanni Resta, May 29 2020

A336672 Unitary barely 3-deficient numbers: numbers m such that usigma(k)/k < usigma(m)/m < 3 for all numbers k < m, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 110670, 182910, 898590, 22851570, 26266170, 45255210, 64124970, 265402410, 1374105810, 1631268870

Offset: 1

Amiram Eldar, Jul 29 2020

Unitary 3-deficient numbers are numbers k such that usigma(k) < 3*k, i.e., numbers that are not in A285615.
The corresponding values of usigma(m)/m are 1, 1.5, 2, 2.4, 2.742..., 2.992..., ...
Are terms squarefree? At some point, do we know that a(n) is divisible by primorial(k) for all n > N(k) for some N(k)? - David A. Corneth, Jul 29 2020
Not all the terms are squarefree. E.g., a(12) = 45255210 is divisible by 11^2.

The unitary version of A307122.

Cf. A034448, A129487, A285615, A302572, A336671.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s = {}; rm = 0; Do[r = usigma[n]/n; If[r < 3 && r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^5}]; s

cp's OEIS Frontend

A034683 Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

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A129485 Odd unitary abundant numbers.

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A302574 Primitive unitary abundant numbers (definition 2): unitary abundant numbers (A034683) having no unitary abundant proper unitary divisor.

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A129468 Unitary abundance of n.

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A097010 Numbers n such that zero is eventually reached when the map x -> A034460(x) is iterated, starting from x = n.

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A302573 Primitive unitary abundant numbers (definition 1): unitary abundant numbers (A034683) all of whose proper unitary divisors are unitary deficient.

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

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