A034683 -id:A034683 - cp's OEIS frontend

A331412 Unitary abundant numbers k such that k + 1 is also unitary abundant.

8857357509, 10783550414, 15197873690, 23620285689, 25537083494, 34736070369, 60326914934, 64139567205, 73969772954, 75776483145, 77509981185, 83968675790, 93092467754, 100012014465, 112236593469, 113606741534, 116519300534, 118905484334, 132584489114, 134889106065

Offset: 1

Amiram Eldar and Giovanni Resta, Jan 18 2020

nonn

Apparently most of the terms are squarefree. Up to 10^13 there are 1150 terms, for only 17 terms k either k or k + 1 is nonsquarefree, and there are no terms k such that both k and k + 1 are nonsquarefree. The first nonsquarefree term is a(32) = 285491549265.

			8857357509 is a term since usigma(8857357509) = 17766604800 > 2 * 8857357509, and usigma(8857357510) = 17851083264 > 2 * 8857357510, where usigma is the sum of unitary divisors function (A034448).

Giovanni Resta, Table of n, a(n) for n = 1..1150 (terms below 10^13)

Cf. A034448, A034683, A292704.

Analogous sequences: A096399 (regular abundant), A283418 (primitive), A318167 (bi-unitary), A327635 (infinitary), A327942 (nonunitary).

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)

A285615 Numbers k such that usigma(k) >= 3*k, where usigma(k) = sum of unitary divisors of k (A034448).

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 79170, 82110, 84630, 85470, 91770, 94710, 99330, 101010, 103530, 108570, 111930, 117390, 122430, 128310, 136290, 140910, 144690, 154770, 161070, 164010, 166530, 168630, 182490, 191730

Offset: 1

Amiram Eldar, Apr 22 2017

nonn

Unitary 3-abundant numbers, correspond to 3-abundant numbers (A023197).
Similarly, the first numbers k such that usigma(k) >= 4*k are 200560490130, 7420738134810, 8222980095330, and 8624101075590. - Giovanni Resta, Apr 23 2017
The least odd term in this sequence is A070826(17) = 961380175077106319535 and the least odd number k such that usigma(k) >= 4*k is A070826(52) = 5.312...*10^95. - Amiram Eldar, Dec 26 2020

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A034448, A023197, A034683, A070826.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Select[Range[100000], usigma[#] >= 3*# &]

isok(k) = sumdivmult(k, d, if(gcd(d, k/d)==1, d)) >= 3*k; \\ Michel Marcus, Dec 26 2020

A380929 Numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

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

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

A302570 Unitary barely abundant numbers: unitary abundant numbers k such that usigma(k)/k < usigma(m)/m for all unitary abundant numbers m < k, where usigma(k) is the sum of the unitary divisors of k (A034448).

Original entry on oeis.org

30, 42, 66, 70, 222, 246, 258, 282, 294, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 726, 750, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434

Offset: 1

Amiram Eldar, Apr 10 2018

nonn

The unitary version of A071927.

			The values of usigma(k)/k are 2.4, 2.285..., 2.181..., 2.057..., 2.054...

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

Cf. A034448, A034683, A071927, A302572.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; seq = {}; r = 3; Do[s = usigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10000}]; seq

A302993 Number of unitary abundant numbers < 10^n.

Original entry on oeis.org

0, 5, 64, 691, 7011, 70030, 699597, 7005450, 70048740, 700321813, 7003128054, 70034216605, 700350142296, 7003426996800, 70033987968599, 700341098675985, 7003416399263312

Offset: 1

Amiram Eldar, Apr 17 2018

Cf. A034448, A034683, A302992.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; uabQ[n_] := usigma[n] > 2 n; c = 0; k = 1; seq={}; Do[While[k < 10^n, If[uabQ[k], c++]; k++]; AppendTo[seq, c], {n, 1, 5}]; seq

Conjecture: Lim_{n->oo} a(n)/10^n = 0.07... is the density of unitary abundant numbers.

a(10)-a(17) from Hiroaki Yamanouchi, Aug 03 2018

A328328 Unitary admirable numbers: numbers k such that there is a proper unitary divisor d of k such that usigma(k) - 2d = 2k, where usigma is the sum of unitary divisors function (A034448).

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, 630, 642, 654, 660, 678, 726, 750, 762, 780, 786, 822, 834, 840, 894, 906, 942, 978, 990, 1002, 1014, 1020, 1038, 1074, 1086

Offset: 1

Amiram Eldar, Oct 12 2019

nonn

Differs from A302574(n) at n >= 30.
Equivalently, numbers that equal to the sum of their proper unitary divisors, with one of them taken with a minus sign.
The unitary version of A111592.
The squarefree terms are also admirable numbers (A111592). The nonsquarefree terms are 150, 294, 420, 630, 660, 726, 750, 780, 840, 990, ...
The unitary abundant numbers (A034683) that are not unitary admirable numbers are: 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 924, 930, 966, ...

			150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper unitary divisors with one of them, 6, taken with a minus sign.

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

Cf. A034448, A077610, A111592, A302574.

Subsequence of A034683 and A290466.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); aQ[n_] := (ab = usigma[n] - 2n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && CoprimeQ[2*n/ab, ab/2]; Select[Range[1086], aQ]

A063846 Numbers k such that sigma(k) - usigma(k) > 2k.

Original entry on oeis.org

1440, 1800, 2160, 2880, 3024, 3600, 4320, 5040, 5400, 5760, 6048, 6480, 7056, 7200, 7560, 7920, 8064, 8640, 9000, 9072, 9360, 9504, 9720, 10080, 10584, 10800, 11088, 11520, 11880, 12096, 12240, 12600, 12960, 13680, 14040, 14112, 14400, 15120

Offset: 1

Jason Earls, Aug 25 2001

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A034448, A048146, A034683, A064597.

u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
j=[]; for(n=1,20000, if(sigma(n)-u(n)>2*n,j=concat(j,n))); j

u(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
{ n=0; for (m=1, 10^9, if(sigma(m) - u(m) > 2*m, write("b063846.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 01 2009

A323341 Numbers k whose unitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

2394, 1452330, 5771934, 5786298, 5800662, 5834178, 5843754, 5858118, 5886846, 5905998, 5920362, 5929938, 5992182, 6035274, 6059214, 6078366, 6087942, 6102306, 6107094, 6121458, 6174126, 6202854, 6207642, 6245946, 6265098, 6274674, 6303402, 6336918, 6360858

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

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

The unitary version of A171641.

Cf. A034448, A034683, A290466, A323342, A323343, A323344.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; seq={}; Do[s=usigma[n]; If[OddQ[s] || s<=2n, Continue[]]; udiv = Select[Divisors[n], GCD[ #, n/# ] == 1 &]; If[Coefficient[Times @@ (1 + x^udiv) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 1500000}]; seq

A333928 Recursive abundant numbers: numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

12, 18, 20, 30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 108, 114, 120, 126, 132, 138, 140, 144, 150, 156, 168, 174, 180, 186, 196, 198, 204, 210, 220, 222, 228, 234, 240, 246, 252, 258, 260, 270, 276, 282, 294, 300, 306, 308, 318, 324, 330, 336, 340, 342, 348

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

			12 is a term since A333926(12) = 28 > 2 * 12.

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

Cf. A333926, A333927.

Analogous sequences: A005101, A034683 (unitary), A064597 (nonunitary), A129575 (exponential), A129656 (infinitary), A292982 (bi-unitary).

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[350], recDivSum[#] > 2*# &]

cp's OEIS Frontend

A331412 Unitary abundant numbers k such that k + 1 is also unitary abundant.

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A129468 Unitary abundance of n.

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A285615 Numbers k such that usigma(k) >= 3*k, where usigma(k) = sum of unitary divisors of k (A034448).

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A380929 Numbers k such that A380845(k) > 2*k.

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A302570 Unitary barely abundant numbers: unitary abundant numbers k such that usigma(k)/k < usigma(m)/m for all unitary abundant numbers m < k, where usigma(k) is the sum of the unitary divisors of k (A034448).

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A302993 Number of unitary abundant numbers < 10^n.

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A328328 Unitary admirable numbers: numbers k such that there is a proper unitary divisor d of k such that usigma(k) - 2d = 2k, where usigma is the sum of unitary divisors function (A034448).

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A063846 Numbers k such that sigma(k) - usigma(k) > 2k.

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